$U(1)_A$ symmetry restoration at finite temperature with mesonic correlators

Using anisotropic FASTSUM Generation 3 ensembles, this study proposes a new method to probe $U(1)_A$ symmetry restoration via mesonic correlators and finds that the symmetry is effectively restored at approximately 320 MeV, a temperature significantly higher than the chiral transition temperature of 180 MeV.

The Gist

Imagine the universe as a giant, complex soup made of tiny particles called quarks. Under normal conditions (like inside a proton), these quarks are stuck together in a very specific way, held by forces that follow strict rules. One of these rules is a kind of "symmetry" called $U(1)_A$. Think of this symmetry like a perfect balance scale: if you swap certain types of particles, the physics should look exactly the same.

However, in our cold, everyday world, this scale is broken. The rules of the quantum world (specifically something called an "anomaly") tip the scale, so the symmetry doesn't exist.

The Big Question:
Scientists have long wondered: What happens if we heat this soup up to extreme temperatures, like just after the Big Bang? Does the scale get fixed? Does the symmetry come back? If it does, when does it happen? Does it happen at the same time the quarks stop sticking together (a moment called the "chiral transition"), or does it happen much later?

The Experiment:
The authors of this paper, a team of physicists, decided to simulate this hot soup on a supercomputer. They used a method called "Lattice QCD," which is like building a 3D grid (a lattice) to represent space and time, and then running a simulation of how quarks behave on that grid.

They used a special type of grid that is "stretched" in the time direction (anisotropic). Imagine a grid made of very thin, tall bricks instead of cubes. This allowed them to take very precise "snapshots" of the particles as they moved through time, giving them a much clearer picture of what was happening.

The Detective Work:
To check if the symmetry was restored, they looked at two specific types of particle pairs:

1. Pions (pseudoscalar mesons)
2. Delta mesons (flavor non-singlet scalar mesons)

If the symmetry is broken (the scale is tipped), these two particles behave very differently. It's like having a red ball and a blue ball that bounce in completely different ways.
If the symmetry is restored (the scale is balanced), these two particles should become identical twins. They should bounce, spin, and interact in exactly the same way.

The Problem with the Tools:
The team used a specific mathematical tool (Wilson-Clover fermions) to run the simulation. While powerful, this tool has a known flaw: it creates "noise" or "artifacts" at very short distances, making it look like the particles are different even when they might be the same. It's like trying to listen to a quiet conversation in a room with a loud fan; the fan makes it hard to tell if the speakers are saying the same thing.

The Solution:
To fix this, the team developed a clever new method. Instead of just looking at the raw data, they:

1. Normalized the data: They adjusted the measurements so that the "loud fan" noise didn't skew the results.
2. Used "Smearing": They blurred the starting and ending points of their measurements slightly. Think of this like putting on a pair of glasses that filters out the static from the radio. This helped them ignore the short-distance noise and focus on the real behavior of the particles.
3. Created a Ratio: They compared the two particles directly. If the ratio is close to zero, they are twins (symmetry restored). If it's far from zero, they are different.

The Results:
They ran the simulation at many different temperatures, from cool to scorching hot.

• At the "Chiral Transition" (approx. 180 MeV): This is the temperature where quarks usually stop being stuck together. The team found that at this point, the two particles were still very different. The symmetry was not restored yet. The scale was still tipped.
• At Higher Temperatures (approx. 320 MeV): As they cranked up the heat even further, the two particles finally started acting like identical twins. The ratio dropped to zero.

The Conclusion:
The paper claims that the $U(1)_A$ symmetry is effectively restored at a temperature of about 320 MeV. This is significantly hotter than the temperature where quarks first become free (180 MeV).

In Simple Terms:
Imagine a party where guests (quarks) are dancing in pairs.

1. At room temperature, the music is broken, and the pairs dance in totally different styles.
2. When the room gets hot (180 degrees), the music stops, and the pairs break up and dance freely, but they still dance in different styles.
3. It isn't until the room gets really hot (320 degrees) that the music fixes itself, and the dancers finally start moving in perfect unison.

The authors conclude that this "perfect unison" (symmetry restoration) happens at a much higher temperature than previously thought by some, and their new method of "smearing" and "ratios" allowed them to see this clearly by filtering out the computer simulation's noise.

Technical Summary

Technical Summary: $U(1)_A$ Symmetry Restoration at Finite Temperature with Mesonic Correlators

Problem Statement
The massless QCD Lagrangian possesses a global symmetry $SU(n_f)_L \times SU(n_f)_R \times U(1)_V \times U(1)_A$. While the chiral $SU(n_f)_L \times SU(n_f)_R$ symmetry is spontaneously broken at zero temperature and restored at a pseudo-critical temperature $T_{pc} \approx 154$ MeV (for physical quark masses), the fate of the $U(1)_A$ symmetry remains an open question. Although explicitly broken by the axial anomaly in the quantized theory, it is hypothesized to be effectively restored at finite temperature, meaning the breaking effects become suppressed. The effective restoration of $U(1)_A$ is expected to manifest as a degeneracy between the pseudoscalar ($\pi$) and flavor non-singlet scalar ($\delta$) mesonic correlation functions. However, current literature has not settled whether this restoration occurs at $T_{pc}$ or at a significantly higher temperature, and previous studies using standard susceptibility differences often suffer from ultraviolet (UV) divergences and short-distance artifacts.

Methodology
This work investigates the effective restoration of $U(1)_A$ symmetry using lattice QCD simulations on anisotropic "Generation 3" FASTSUM ensembles. The study employs Wilson-Clover fermions with a pion mass of $m_\pi \approx 378$ MeV and an anisotropy ratio $\xi = a_s/a_\tau \approx 7$. The high anisotropy allows for a fine temperature resolution using the fixed-scale approach, spanning $T \in [50, 534]$ MeV.

To address the limitations of standard susceptibility measurements ($\chi_\pi - \chi_\delta$), which are sensitive to operator overlaps and UV artifacts, the authors develop and compare three specific methods based on mesonic correlators:

1. Standard Susceptibility Difference: Calculating the sum of the difference between pseudoscalar and scalar correlators. This method is found to be insufficient as the difference does not vanish even at high temperatures due to differing operator overlaps.
2. Normalized and Cutoff Susceptibility: Introducing a short-distance temporal cutoff ($\tau_{min}$) and normalizing correlators at their mid-point ($N_\tau/2$). This removes dependence on operator overlaps and suppresses short-distance artifacts.
3. Integrated Ratio of Smeared Correlators: Constructing a ratio $R(\tau)$ of the difference to the sum of mid-point normalized correlators. To further mitigate artifacts arising from the Wilson term (specifically the upward curvature at lattice edges), the authors employ Gaussian smearing on the source and sink. They systematically tune the smearing radius ($r_{RMS}$) to eliminate short-distance artifacts while preserving the ground state dominance. Finally, they define an integrated ratio $R(\tau_{min}, N_\tau/2; T)$ by summing the ratio over the temporal extent, weighted by uncertainties.

Key Results

• Standard vs. Improved Methods: The standard susceptibility difference ($\chi_\pi - \chi_\delta$) remains non-zero across the temperature range, failing to indicate degeneracy. In contrast, the normalized and cutoff method shows the difference approaching zero at $T \gtrsim 225$ MeV.
• Smearing Optimization: Analysis of the ratio $R(\tau)$ with varying smearing radii reveals that unsmeared correlators exhibit significant artifacts at the lattice edges. A smearing radius of $r_{RMS} = 2.45$ (corresponding to $\kappa=1.96, n=6$) is identified as optimal, effectively removing the upward curvature while maintaining agreement with local correlators near the mid-point.
• Restoration Temperature: Using the integrated ratio with optimally smeared correlators, the authors determine the temperature at which the $\pi$ and $\delta$ channels become degenerate. The data indicates that the correlators are not degenerate at the chiral transition temperature ($T_{pc} \sim 180$ MeV). Instead, the integrated ratio crosses zero at $T_{U(1)_A} = 317(4)$ MeV (statistical uncertainty only).

Significance and Claims
The paper claims to provide a robust determination that the effective restoration of $U(1)_A$ symmetry occurs well above the chiral transition temperature ($T_{U(1)_A} \approx 320$ MeV vs. $T_{pc} \approx 180$ MeV) for the chosen Wilson-Clover fermion setup. The authors emphasize that their methodology, particularly the use of anisotropic ensembles combined with smeared correlators and an integrated ratio, offers a novel way to examine susceptibilities before summation, thereby isolating infrared effects relevant to symmetry restoration while controlling systematic errors from the Wilson term and UV artifacts.

The work confirms the lack of $U(1)_A$ restoration at $T_{pc}$ and suggests the existence of a high-temperature phase where this symmetry is effectively restored. The authors note that their determined temperature is close to transition temperatures found via center vortex studies, hinting at a possible third high-temperature phase of QCD matter. Future work is planned to include Generation 2 and 2L ensembles to quantify systematic uncertainties related to pion mass and anisotropy, and to apply these methods to the study of chiral spin symmetry.