Topological susceptibility and QCD phase transition with 2+1 flavor Möbius domain wall fermion at finite temperature

This paper presents lattice QCD results for the topological susceptibility, chiral condensate, and disconnected susceptibility using 2+1 flavor Möbius domain wall fermions at the physical point across a finite temperature range of approximately 140 MeV to 500 MeV.

The Gist

Imagine the universe as a giant, invisible ocean made of pure energy. Deep inside this ocean, there are tiny, fundamental particles called quarks that dance together to form protons and neutrons (the building blocks of everything we see). The rules that govern how these quarks dance are called Quantum Chromodynamics (QCD).

This paper is like a report from a team of scientists (the JLQCD collaboration) who built a giant, digital aquarium to watch these quarks dance. They wanted to see what happens when they heat up the water until the quarks start behaving differently.

Here is the breakdown of their experiment in simple terms:

1. The Challenge: The "Pixelated" Problem

To simulate the universe on a computer, scientists have to break space and time into tiny blocks, like pixels on a screen. This is called a lattice.

• The Problem: If the pixels are too big (a "coarse" lattice), the picture gets blurry and distorted. The scientists found that one specific measurement, called Topological Susceptibility, is very sensitive to these blurry pixels. It's like trying to count the ripples on a pond, but your camera is so low-resolution that you can't tell if a ripple is real or just a glitch in the image.
• The Solution: They used a special type of mathematical rule for their quarks called Möbius Domain Wall Fermions. Think of this as a super-high-definition camera lens that keeps the "chiral symmetry" (a fundamental rule of how quarks spin) perfectly intact, even when the pixels aren't perfect. This helps them get a much clearer picture than previous attempts.

2. The Experiment: Heating Up the Ocean

The team simulated the quark ocean at different temperatures, ranging from a cozy 140 degrees (in particle physics terms) to a scorching 500 degrees.

• The Goal: They wanted to find the exact moment the "phase transition" happens. Imagine ice melting into water. At a certain temperature, the quarks stop being stuck in their little groups (protons/neutrons) and turn into a free-flowing soup called the Quark-Gluon Plasma.
• The Result: They found this "melting point" (called the pseudo-critical temperature) happens around 155 MeV (about 1.8 trillion degrees Celsius). This matches what other teams found using different methods, confirming their "camera lens" is working correctly.

3. The Mystery of the "Topological Charge"

This is the most complex part of the paper, but here is the analogy:

• The Knots: Imagine the quark ocean isn't just smooth water; it's full of invisible knots and twists. These knots are called "topological charges."
• The Susceptibility: "Topological Susceptibility" is a measure of how many of these knots exist and how easily they wiggle around.
• Why it Matters: These knots are crucial for understanding Axions, a hypothetical particle that might make up Dark Matter (the invisible stuff holding galaxies together). If we know how the knots behave at high heat, we can figure out how much Dark Matter exists in the universe.
• The Discovery:
  • At lower temperatures, the ocean is full of knots.
  • As they heated the ocean to 500 degrees, the knots started to disappear. At the hottest point, the ocean became so smooth that zero knots were found.
  • The "Freezing" Issue: At high temperatures, the computer simulation gets "stuck" in one state (like a video game character frozen in a wall). The scientists had to use clever tricks to make sure they were actually seeing the knots disappear and not just getting stuck in a glitch.

4. The Takeaway

The scientists successfully built a high-resolution digital model of the quark ocean.

• They confirmed the temperature at which normal matter turns into plasma.
• They showed that their special "Möbius" camera lens reduces the "pixelation errors" better than other methods.
• They provided new, cleaner data on how the "knots" in the universe behave as things get hotter.

In a nutshell: They used a super-advanced digital microscope to watch the universe's fundamental building blocks melt. Their results help us understand the early moments of the Big Bang and hunt for the mysterious Dark Matter that fills our cosmos. The simulation is still running, and they plan to get even more data to make the picture crystal clear.

Technical Summary

1. Problem Statement

The paper addresses two critical challenges in Lattice Quantum Chromodynamics (QCD) simulations at finite temperature:

• Chiral Symmetry Preservation: Accurately simulating the restoration of chiral symmetry requires a fermion discretization that minimizes explicit chiral symmetry breaking. Standard actions often suffer from large additive mass renormalizations and discretization errors.
• Topological Susceptibility ($\chi_{top}$) Discretization Errors: $\chi_{top}$ is a fundamental quantity characterizing the QCD vacuum and is crucial for axion dark matter scenarios. However, it is highly sensitive to fermion discretization choices and suffers from large systematic errors, particularly at high temperatures. Previous studies using different fermion formulations (e.g., HISQ, Domain Wall) have shown significant discrepancies in high-temperature results. Additionally, "topology freezing" at high temperatures makes sampling different topological sectors difficult.

2. Methodology

The authors employed 2+1 flavor Möbius Domain Wall Fermions (MDWF) to perform simulations at the physical point (realistic quark masses).

• Action and Setup:

  • Fermion Action: Möbius domain wall fermions with a scale factor of 2.
  • Gauge Action: Tree-level Symanzik-improved gauge action.
  • Smearing: 3 steps of stout smearing ($\rho=0.1$) applied to the gauge field within the fermion action to improve locality and reduce noise.
  • Quark Masses: Set to the physical point along the line of constant physics. The renormalized light quark mass ($m_l^R$) is set to $1/27.4$ of the strange quark mass ($m_s^R$) at $\mu=2$ GeV. Residual mass corrections were carefully subtracted.
  • Fifth Dimension: Fixed extent $L_s = 12$ for all ensembles.

• Lattice Parameters:

  • Temperatures: Ranged from $\sim 130$ MeV to $500$ MeV.
  • Temporal Lattice Sizes ($N_t$):
    • $N_t = 12$ and $16$ for the chiral condensate and susceptibility (covering 140–250 MeV).
    • $N_t = 10$ for topological susceptibility measurements extending up to 500 MeV.
  • Volumes: Multiple spatial volumes were used to check finite-size effects (e.g., $36^3 \times 12$, $48^3 \times 12$, $48^3 \times 16$, $64^3 \times 16$, $40^3 \times 10$).
  • Computing: Simulations were performed on the Fugaku supercomputer using the Grid library for generation and measurement.

• Observables:

  • Chiral Condensate ($\langle \bar{\psi}\psi \rangle$): Renormalized by removing additive divergence (proportional to $1/a^2$) and multiplicative divergence (via $Z_m^{MS}$).
  • Disconnected Susceptibility ($\chi_{disc}$): Calculated as the variance of the condensate to cancel additive divergences.
  • Topological Charge ($Q$): Measured using a clover discretization after applying Wilson flow (flow time $t=5.0$ in lattice units). $\chi_{top} = \langle Q^2 \rangle / V$.

3. Key Contributions

• Systematic Control of Residual Mass: The study demonstrates effective control over the residual mass inherent to finite $L_s$ domain wall fermions, ensuring the chiral condensate is well-defined in the continuum limit.
• High-Precision Chiral Transition: Provided a precise determination of the pseudo-critical temperature ($T_c$) using MDWF, a formulation known for excellent chiral symmetry properties.
• Discretization Error Analysis: Offered a comparative analysis of topological susceptibility across different lattice spacings ($N_t=10, 12, 16$) to quantify discretization errors, a known issue where MDWF results differ from other formulations like HISQ.
• Topology Freezing Investigation: Investigated the behavior of topological charge distributions at high temperatures ($T \ge 400$ MeV), where topology freezing becomes severe.

4. Key Results

A. Chiral Condensate and Susceptibility

• Chiral Condensate: As expected, the condensate decreases with temperature, approaching zero as chiral symmetry is restored.
• Pseudo-critical Temperature ($T_c$): The peak of the disconnected chiral susceptibility was located in the range 153–157 MeV.
  • This result is consistent with continuum limit results from HISQ ($156.5 \pm 1.5$ MeV) and stout actions ($158.0 \pm 0.6$ MeV).
  • It also aligns with previous MDWF studies ($N_t=8$), validating the consistency of the MDWF approach.
• Volume/Discretization Effects: No significant volume dependence was observed, except for the $q3$ series ($64^3 \times 16$) which had lower statistics.

B. Topological Susceptibility ($\chi_{top}$)

• Discretization Effects: Finer lattices ($N_t=16$) yielded systematically smaller values of $\chi_{top}$ compared to coarser lattices ($N_t=12$).
  • At low temperatures ($T \le 150$ MeV), the $N_t=12$ results overshoot the $T=0$ continuum value, indicating sizable discretization errors at this resolution.
  • The $N_t=16$ results are significantly closer to the continuum limit values reported by other collaborations (e.g., HISQ results at $T=140$ MeV).
• Continuum Extrapolation: At $T=250$ MeV, data from $N_t=10, 12, 16$ suggest that a linear extrapolation in $a^2$ is reasonable.
• High Temperature Behavior ($T \ge 400$ MeV):
  • Topology freezing was observed. At $T=500$ MeV, only configurations with $Q=0$ were observed.
  • The distribution of topological charge $Q$ becomes concentrated at zero as temperature increases.
• Finite Volume Effects: Results with aspect ratio 3 were slightly smaller than those with aspect ratio 4, though statistical fluctuations prevent a definitive conclusion on volume dependence.

5. Significance and Outlook

• Validation of MDWF: The results confirm that Möbius domain wall fermions provide a robust framework for studying the QCD phase transition with minimal chiral symmetry breaking artifacts. The derived $T_c$ is consistent with other state-of-the-art formulations.
• Discretization Sensitivity: The study highlights that topological susceptibility is extremely sensitive to lattice spacing. The discrepancy between $N_t=12$ and $N_t=16$ results underscores the necessity of using fine lattices ($N_t \ge 16$) or performing rigorous continuum extrapolations for accurate $\chi_{top}$ determination.
• Axion Physics: By providing more reliable constraints on $\chi_{top}$ at finite temperature, this work contributes to refining predictions for the axion dark matter abundance, which depends heavily on the temperature dependence of $\chi_{top}$.
• Future Work: The authors note that the simulation is ongoing. Future work will focus on increasing statistics (especially for the large $64^3 \times 16$ volume), developing analysis techniques to overcome topology freezing at $T \ge 400$ MeV (beyond simple global charge measurement), and performing a full continuum limit analysis.