On the effective restoration of $U(1)_A$ symmetry at finite temperature

Using anisotropic lattice QCD ensembles with Wilson-clover fermions, the study finds evidence for the effective restoration of $U(1)_A$ symmetry at a temperature of $319(22)$ MeV, which is significantly higher than the chiral crossover temperature.

The Gist

Imagine the universe as a giant, cosmic soup. For most of its history, this soup has been cool enough that the ingredients—quarks and gluons—stick together to form "particles" like protons and neutrons (the building blocks of everything we see). But if you heat this soup up to temperatures trillions of degrees hotter than the center of the sun, the ingredients melt apart into a free-flowing plasma called the Quark-Gluon Plasma.

This paper is a scientific investigation into what happens to the "rules of the game" (symmetries) when the universe is in this super-hot state. Specifically, the authors are looking for the moment when a specific rule, called U(1)A symmetry, gets "switched back on."

Here is a breakdown of the story, using simple analogies:

1. The Broken Mirror (The Problem)

In the cold, everyday world, nature has a set of rules. One of these rules is Chiral Symmetry. Imagine a pair of gloves: a left-handed one and a right-handed one. In the cold universe, these gloves are distinct and behave differently. This is why particles like the pion (a type of meson) are light, while their "partners" are heavy.

However, there is a second, more subtle rule called U(1)A symmetry. In the cold universe, this rule is broken by a "glitch" in the system (called an anomaly). Because of this glitch, a particle called the $\eta'$ meson is much heavier than it "should" be. It's like having a mirror that is cracked; the reflection doesn't match the object.

2. The Great Melting (The Hypothesis)

Physicists have long suspected that if you heat this cosmic soup up enough, the "glitch" might disappear. If the U(1)A symmetry is restored, the mirror becomes whole again. The heavy and light particles that were previously different would suddenly become identical twins (degenerate).

The big question is: At what temperature does this happen?

• We know the "Chiral Symmetry" (the gloves) gets restored around 154 MeV (about 1.8 trillion degrees).
• But does the U(1)A symmetry (the mirror) get fixed at the same time, or does it take even more heat?

3. The Experiment (The Method)

To find the answer, the authors didn't use a real pot of soup (it's too hot and dangerous!). Instead, they used a supercomputer to simulate the universe on a grid. This is called Lattice QCD.

• The Grid: Imagine a 3D chessboard, but with an extra dimension for time.
• The Challenge: The "glitch" (anomaly) is very sensitive to the size of the squares on the chessboard. If the squares are too big, the simulation gets messy and gives wrong answers.
• The Solution: The team used a special "anisotropic" grid. Think of it like a loaf of bread where the slices are very thin (in the time direction) but the width of the loaf is normal. This gives them a super-high-resolution "slow-motion" view of the particles as they vibrate, allowing them to see the subtle effects of the symmetry restoration clearly.

They ran simulations on three different "generations" of these grids, with the newest one (Generation 3) being the most detailed and covering the widest range of temperatures.

4. The Discovery (The Result)

The team looked at how different types of particles (specifically, "pseudoscalar" and "scalar" mesons) behaved as they heated up the simulation.

• At lower temperatures: The particles were distinct. The "mirror" was still cracked. The difference between them was clear.
• At higher temperatures: As they cranked up the heat past a certain point, the difference between the particles vanished. They became identical twins.

The Verdict:
The U(1)A symmetry does not get restored at the same time as the chiral symmetry. It waits until the soup is much hotter.

• Chiral Symmetry restores: ~154 MeV.
• U(1)A Symmetry restores: ~319 MeV (roughly double the temperature).

5. Why This Matters

This finding is like finding a hidden layer in the universe's operating system.

1. Two Steps, Not One: It suggests that the transition from normal matter to the Quark-Gluon Plasma isn't a single "explosion" of change, but a two-step process. First, the particles melt (chiral symmetry), and then, at a much higher temperature, the deep quantum rules (U(1)A) finally reset.
2. The Early Universe: This helps cosmologists understand what the universe looked like microseconds after the Big Bang. It tells us there was a specific "phase" of the universe where the soup was hot enough to melt particles, but not hot enough to fix the U(1)A glitch.
3. The "In-Between" Phase: This discovery hints at a mysterious intermediate state of matter that exists between the two transition temperatures, which might have its own unique properties.

Summary in a Nutshell

The authors used a high-resolution computer simulation to prove that the universe's "rules of symmetry" don't all break and fix at the same time. While the basic structure of matter changes at a certain temperature, a deeper, more subtle rule (U(1)A) stays broken until the temperature is nearly twice as high. They found the "switch" for this rule flips at 319 MeV, a discovery that refines our map of the universe's most extreme conditions.

Technical Summary

1. Problem Statement

The paper addresses a fundamental open question in Quantum Chromodynamics (QCD): the temperature at which the $U(1)_A$ axial symmetry, explicitly broken by the axial anomaly in the vacuum, is effectively restored at finite temperature.

• Context: While chiral symmetry ($SU(N_f)_L \times SU(N_f)_R$) is known to be restored at the chiral crossover temperature ($T_c \approx 154$ MeV for physical quark masses), the fate of $U(1)_A$ is less certain.
• Significance: The restoration of $U(1)_A$ is crucial for determining the order of the chiral phase transition in the chiral limit and understanding the structure of the QCD phase diagram.
• Previous Discrepancies: Lattice QCD studies have yielded conflicting results. Some suggest restoration occurs near $T_c$, while others indicate it happens at much higher temperatures. A major challenge has been the use of lattice fermion formulations that either break chiral symmetry explicitly (like Wilson fermions) or suffer from lattice artifacts at short distances (like Domain Wall fermions), making the identification of the degeneracy between chiral partners difficult.

2. Methodology

The authors utilize anisotropic lattice QCD simulations to overcome previous limitations regarding temporal resolution and chiral symmetry breaking effects.

• Lattice Ensembles: The study uses "Fastsum" collaboration ensembles (Generations 2, 2L, and 3) with $N_f = 2+1$ flavors.
  • Action: Symanzik improved anisotropic gauge action with mean-field-improved Wilson-clover fermions and stout-smeared links.
  • Anisotropy ($\xi = a_s/a_\tau$): The ensembles feature high anisotropy, particularly Generation 3 with $\xi \approx 7.0$, allowing for a much finer temporal lattice spacing ($a_\tau$) compared to spatial spacing ($a_s$). This provides unprecedented resolution in the Euclidean time direction.
  • Temperatures: A wide range of temperatures is covered, from below $T_c$ up to $\sim 760$ MeV.
• Observables:
  • The study focuses on the degeneracy between the flavour non-singlet pseudoscalar ($\pi$) and scalar ($\delta$ or $a_0$) channels. Effective restoration implies these channels become indistinguishable.
  • Correlation Functions: Euclidean correlation functions $G(\tau, T)$ are computed.
  • Susceptibilities: The integrated susceptibilities $\chi$ are analyzed. However, due to the explicit chiral symmetry breaking of Wilson fermions, the standard susceptibility difference $\chi_\pi - \chi_\delta$ suffers from contact term contributions and renormalization issues.
• Mitigation Strategies:
  1. Smearing: The authors use "smeared" operators (Gaussian smearing with $r_{RMS} = 2.45$) to reduce overlap with excited states and suppress short-distance lattice artifacts associated with the Wilson mass term.
  2. Normalization: To cancel renormalization factors and isolate infrared physics, they define a normalized ratio:
     $$R_{\pi-\delta}(\tau, T) = \frac{\tilde{G}_\pi(\tau, T) - \tilde{G}_\delta(\tau, T)}{\tilde{G}_\pi(\tau, T) + \tilde{G}_\delta(\tau, T)}$$
     where $\tilde{G}$ is the correlator normalized at the midpoint $\tau = N_\tau/2$.
  3. Temporal Cutoff ($\tau_{min}$): The analysis excludes the short-distance region ($\tau < \tau_{min}$) where Wilson artifacts dominate. Systematic uncertainties are estimated by varying $\tau_{min}$.

3. Key Contributions

• First Large-Scale Wilson Study: This is the first extensive study of $U(1)_A$ restoration using Wilson-clover fermions on highly anisotropic lattices. It demonstrates that despite the explicit breaking of chiral symmetry by the Wilson term, effective restoration can be reliably extracted if short-distance artifacts are carefully controlled via smearing and temporal cutoffs.
• High-Resolution Temporal Data: The use of Generation 3 ensembles ($N_\tau$ up to 256, $\xi \approx 7$) provides a temporal resolution previously unattainable in this context, allowing for a precise determination of the degeneracy point.
• Systematic Control: The paper rigorously addresses the systematic errors inherent to Wilson fermions by comparing local vs. smeared operators and validating results against non-interacting fermion models to isolate lattice artifacts.

4. Results

• Degeneracy Observation:
  • At low temperatures ($T < 200$ MeV), the ratio $R_{\pi-\delta}$ shows a distinct inverted parabolic shape, indicating a mass splitting between the $\pi$ and $\delta$ channels ($m_\delta > m_\pi$).
  • As temperature increases, the curvature of this ratio decreases.
  • At the highest temperatures ($T \gtrsim 300$ MeV), the ratio becomes consistent with zero across the temporal extent, indicating the degeneracy of the pseudoscalar and scalar channels.
• Restoration Temperature ($T_{U(1)A}$):
  • By interpolating the central values of $R_{\pi-\delta}(T)$ using a cubic spline for Generation 3 data, the authors determine the temperature where the ratio crosses zero.
  • Result: $T_{U(1)A} = 319(22)$ MeV.
  • This temperature is significantly higher than the chiral crossover temperature ($T_c \approx 154$ MeV).
• Comparison with Generations 2/2L: Results from Generations 2 and 2L (lower anisotropy) approach zero but do not exhibit a clear crossing within errors, highlighting the necessity of the finer temporal resolution provided by Generation 3.
• Chiral Symmetry: A parallel analysis for $SU(2)_L \times SU(2)_R$ restoration (using vector/axial-vector correlators) confirms a lower transition temperature, consistent with $T_c$, reinforcing the separation of the two symmetry restoration scales.

5. Significance

• Resolution of Ambiguity: The results strongly suggest that $U(1)_A$ symmetry is not restored at the chiral crossover ($T_c$) but rather at a much higher temperature ($T \approx 2 T_c$). This resolves the ambiguity in previous literature where some studies suggested restoration near $T_c$.
• Phase Diagram Structure: The findings imply a two-stage transition scenario for QCD with physical quark masses:
  1. A crossover at $T_c \approx 154$ MeV where chiral symmetry is restored.
  2. A subsequent effective restoration of $U(1)_A$ at $T \approx 319$ MeV.
• Theoretical Implications: The temperature $T_{U(1)A} \approx 320$ MeV coincides with the regime where:
  • Topological susceptibility follows a power-law scaling consistent with the Dilute Instanton Gas Approximation (DIGA).
  • Emergent Chiral-Spin (CS) symmetry has been proposed.
  • Deconfinement transitions based on center vortices have been identified.
• Methodological Validation: The study validates the use of Wilson fermions for studying symmetry restoration, provided that high anisotropy and smearing techniques are employed to mitigate lattice artifacts. This opens the door for future high-precision studies using Wilson-type actions, which are computationally more efficient than chirally symmetric formulations like overlap fermions.