RG-Invariant Symmetry Ratio for QCD: A Study of $U(1)_A$ and Chiral Symmetry Restoration

Using an RG-invariant symmetry ratio on $N_f=2+1+1$ lattice QCD data, this study demonstrates that while a hierarchy of symmetry-breaking strengths exists at finite lattice spacing, the continuum extrapolation reveals that $SU(2)_L \times SU(2)_R$ and $U(1)_A$ chiral and axial symmetry restoration in the nonsinglet sector converge closely near the chiral crossover, supporting a two-stage restoration scenario where full symmetry recovery requires higher temperatures to suppress topological fluctuations.

The Gist

Imagine the universe is built from tiny, invisible Lego bricks called quarks. These bricks stick together to form protons and neutrons, which make up everything we see. The "glue" that holds them together is a force called the Strong Interaction, governed by a set of rules known as Quantum Chromodynamics (QCD).

In the cold, quiet vacuum of space, these rules are broken in specific ways. It's like a perfectly symmetrical snowflake that has been melted and refrozen into a lopsided ice cube. This "broken symmetry" is why particles have mass and why the universe looks the way it does.

But what happens if you heat this ice cube until it melts into a super-hot soup? This is what happens in the early universe or inside heavy-ion collision experiments (like smashing gold atoms together). Scientists want to know: At what temperature do the rules get "fixed" again? Do all the broken symmetries snap back into place at the exact same moment, or do they fix themselves one by one?

The Problem: The "Pixelated" Photo

To answer this, physicists use supercomputers to simulate the universe on a grid (a lattice). Think of this grid like a digital photo. If the photo is low-resolution (large "pixels"), things look blurry and distorted. You might see a difference between two objects that isn't actually there, just because of the pixelation.

Previous studies were like looking at low-resolution photos. They suggested that one type of symmetry (let's call it Symmetry A) fixes itself at a lower temperature, while another (Symmetry B) waits until it gets much hotter. But because the "pixels" were too big, the scientists couldn't be sure if this was real or just a glitch in the picture.

The New Tool: The "Symmetry Ruler"

In this paper, the authors introduce a new, super-precise tool called $\kappa_{AB}$ (the symmetry strength parameter).

Imagine you have two twins, Alice and Bob.

• If the rules are broken, Alice and Bob look different (maybe one is tall, one is short).
• If the rules are restored, they look identical.

Old methods tried to measure their height at a single moment, which could be noisy or misleading. The new tool, $\kappa_{AB}$, is like a smart ruler that measures the entire difference between Alice and Bob over time, averages it out, and gives a single number between 0 and 1.

• 0 means they are perfect twins (Symmetry is restored).
• 1 means they are totally different (Symmetry is broken).

Crucially, this ruler is "RG-invariant." In plain English, this means it doesn't care about the resolution of the photo. Whether you look at a blurry low-res image or a crystal-clear 8K image, the ruler gives you the same true answer.

The Experiment: Heating Up the Soup

The team ran massive simulations using three different "resolutions" (lattice spacings) and heated the system from 164 MeV to 385 MeV (roughly 2 trillion degrees Kelvin). They looked at three different pairs of "twins" (particles):

1. The Scalar-Pseudoscalar pair ($\kappa_{PS}$): Sensitive to the "U(1)A" symmetry.
2. The Vector-Axial pair ($\kappa_{VA}$): Sensitive to the "Chiral" symmetry.
3. The Tensor-Axial-Tensor pair ($\kappa_{TX}$): Another way to check the "U(1)A" symmetry.

The Results: The Illusion of Hierarchy

At low resolution (finite lattice spacing):
When they looked at the blurry, low-res data, they saw a clear order:

• The Scalar-Pseudoscalar twins looked very different.
• The Vector-Axial twins looked somewhat different.
• The Tensor twins looked the most similar.
  It looked like the symmetries were fixing themselves in a specific order, one after another.

At high resolution (Continuum Limit):
When they used their new ruler to correct for the "pixelation" and extrapolated to the perfect, infinite-resolution limit, the order collapsed.

Suddenly, all three pairs of twins looked statistically identical. The differences between them vanished.

• The Conclusion: The "hierarchy" was an illusion caused by the low-resolution grid. In reality, the Chiral symmetry and the U(1)A symmetry in the "nonsinglet" sector (the parts of the soup made of connected quarks) restore themselves at almost the exact same temperature.

The Twist: The Two-Stage Restoration

So, does everything fix itself at once? Not quite. The authors propose a Two-Stage Restoration scenario, like a house being renovated:

1. Stage 1 (The Main Room): Around 156 MeV (the temperature of the crossover), the "nonsinglet" parts of the symmetries get fixed. The twins we measured (Alice, Bob, etc.) become identical. The "glue" holding the specific particle pairs together stops breaking the rules.
2. Stage 2 (The Basement): However, there is a "basement" of the system involving topological fluctuations (weird, knotted structures in the fabric of space-time). These are like hidden, stubborn knots that don't untie until the temperature gets much higher (Stage 2).
   • Until these knots untie, the "singlet" particles (the ones connected to these knots) remain different.
   • So, while the main symmetries look restored to our "ruler," the full, perfect symmetry of the universe only returns at a much higher temperature when these topological knots finally disappear.

Why This Matters

This paper is a big deal because it settles a long-standing debate. It proves that for the parts of the universe we can easily see (the quark-connected parts), the symmetries return together, not in a staggered line. It also gives us a clear roadmap for the future: to understand the full restoration, we need to look deeper into the "basement" (the topological knots) at even higher temperatures.

In short: The authors built a better ruler, took a clearer picture, and discovered that the universe's symmetries snap back into place much more simultaneously than we thought—though a few stubborn knots remain until it gets even hotter.

Technical Summary

1. Problem Statement

The paper addresses a long-standing open question in Quantum Chromodynamics (QCD): the relative restoration scales of two global symmetries at finite temperature:

• Chiral Symmetry ($SU(2)_L \times SU(2)_R$): Spontaneously broken in the vacuum, its restoration marks the transition to the Quark-Gluon Plasma (QGP) at $T_c \approx 156$ MeV.
• Axial Symmetry ($U(1)_A$): Explicitly broken by the axial anomaly (related to topological fluctuations).

The central debate is whether $U(1)_A$ symmetry is restored simultaneously with chiral symmetry at $T_c$, or if it remains broken until a significantly higher temperature ($T_1 > T_c$) where topological fluctuations are suppressed. Previous lattice QCD studies have yielded conflicting results, often due to:

1. Lattice Artifacts: Discretization effects that mimic symmetry breaking.
2. Observable Ambiguity: Traditional probes (e.g., hadron masses or single-time correlators) are sensitive to analysis choices and renormalization schemes.
3. Channel Dependence: Different studies focused on different symmetry-breaking channels, leading to inconsistent hierarchies.

2. Methodology

The authors introduce a novel, robust framework to resolve these issues:

A. The RG-Invariant Symmetry Ratio ($\kappa_{AB}$)

Instead of comparing raw correlation functions, the authors define a Renormalization-Group (RG) invariant observable, the symmetry strength parameter $\kappa_{AB}$:
$$\kappa_{AB}(T) = \frac{\chi_A(T) - \chi_B(T)}{\chi_A(T) + \chi_B(T)}$$
Where $\chi_A$ and $\chi_B$ are regularized susceptibilities (integrated spectral weights of Euclidean correlators) for two operators $A$ and $B$ that are symmetry partners.

• RG Invariance: Because the renormalization constants ($Z_A = Z_B$) cancel out for symmetry partners in chirally symmetric formulations, $\kappa_{AB}$ is independent of the renormalization scheme and scale.
• Regularization: The susceptibility is defined as the difference between the value at temperature $T$ and a reference temperature $T_r$ (where symmetry is restored), canceling additive UV divergences.
• Interpretation: $\kappa_{AB} = 0$ implies exact degeneracy (symmetry restoration); $\kappa_{AB} \to 1$ implies maximal breaking.

B. Lattice Setup

• Action: $N_f = 2+1+1$ optimal Domain-Wall Fermions (preserving chiral symmetry to high precision) at the physical point.
• Parameters: Three lattice spacings ($a \approx 0.075, 0.069, 0.064$ fm) and twelve temperatures ranging from 164 MeV to 385 MeV.
• Channels Studied: Three independent symmetry-breaking channels in the nonsinglet (quark-connected) sector:
  1. $\kappa_{PS}$ (Scalar-Pseudoscalar): Probes $U(1)_A$ breaking via the $\pi$-$\delta$ system.
  2. $\kappa_{VA}$ (Vector-Axial-Vector): Probes $SU(2)_L \times SU(2)_R$ breaking via the $\rho$-$a_1$ system.
  3. $\kappa_{TX}$ (Tensor-Axial-Tensor): Probes $U(1)_A$ breaking via the $\rho_T$-$b_1$ system (using a different Dirac structure than $\kappa_{PS}$).

3. Key Contributions

1. New Observable: The introduction of $\kappa_{AB}$ as a universal, model-independent, and RG-invariant diagnostic for symmetry restoration.
2. Continuum Extrapolation: The first controlled continuum extrapolation ($a \to 0$) of symmetry restoration ratios across multiple channels using chirally symmetric fermions.
3. Resolution of Discrepancies: Demonstrating that apparent hierarchies in symmetry breaking observed at finite lattice spacing are artifacts that vanish in the continuum limit.

4. Results

Finite Lattice Spacing ($a > 0$)

At non-zero lattice spacing, a clear hierarchy is observed:
$$\kappa_{PS} > \kappa_{VA} > \kappa_{TX}$$
This suggests that $U(1)_A$ breaking (measured by $\kappa_{PS}$) is stronger than chiral breaking ($\kappa_{VA}$), and the tensor channel ($\kappa_{TX}$) shows the weakest breaking. If taken at face value, this would imply $U(1)_A$ restoration occurs at a much higher temperature than chiral restoration.

Continuum Limit ($a \to 0$)

Upon performing a controlled continuum extrapolation using a power-law ansatz:

• The hierarchy collapses.
• All three parameters ($\kappa_{PS}, \kappa_{VA}, \kappa_{TX}$) become statistically indistinguishable within the precision of the study across the entire temperature range (164–385 MeV).
• The values converge to zero near the chiral crossover temperature ($T_c \approx 156$ MeV), indicating that the effective restoration scales for both $SU(2)_L \times SU(2)_R$ and $U(1)_A$ in the nonsinglet sector are nearly identical.

5. Significance and Physical Interpretation

The Two-Stage Restoration Scenario

The results support a two-stage hierarchical restoration picture for QCD:

1. Stage 1 (Nonsinglet Restoration, $T \sim T_c$):
   • Quark-connected correlators (nonsinglet channels) for both chiral and axial symmetries become degenerate.
   • The $U(1)_A$ anomaly effects on these specific channels are suppressed by thermal screening.
   • This is confirmed by the vanishing of $\kappa_{PS}, \kappa_{VA}, \kappa_{TX}$.
2. Stage 2 (Full Restoration, $T \gg T_c$):
   • Full effective restoration, including flavor-singlet channels (e.g., $\sigma, \eta, \omega, h_1$), requires the suppression of topological fluctuations (Topological Susceptibility $\chi_t \to 0$).
   • Since $\chi_t$ remains finite well above $T_c$ (up to $2T_c$ or higher), the singlet channels remain split.
   • Therefore, while the nonsinglet sector restores near $T_c$, the full $U(1)_A$ symmetry (including singlets) is only restored at a significantly higher temperature $T_s^c$.

Implications

• Model Independence: The finding that $\kappa_{PS} \approx \kappa_{VA}$ in the continuum limit places stringent constraints on effective models that predict a large separation between chiral and axial restoration temperatures.
• Lattice Artifacts: The study highlights that finite-lattice-spacing effects can create misleading hierarchies in symmetry breaking, emphasizing the necessity of continuum extrapolations.
• Emergent Symmetries: The results align with observations of emergent approximate symmetries (like $SU(2)_{CS}$) in the intermediate temperature window ($T_c < T < 2T_c$), where chromoelectric interactions dominate before full deconfinement.

Conclusion

The paper establishes that in the nonsinglet sector of QCD, chiral ($SU(2)_L \times SU(2)_R$) and axial ($U(1)_A$) symmetries effectively restore concurrently near the chiral crossover temperature. The apparent delay in $U(1)_A$ restoration observed in some previous studies is largely an artifact of finite lattice spacing or a misinterpretation of singlet vs. nonsinglet dynamics. Full restoration requires the suppression of topological fluctuations, which occurs at a higher temperature scale.