Revisiting the sphaleron and axion production rates in QCD at high temperatures

This paper presents new lattice results for sphaleron and non-perturbative thermal axion production rates in QCD across a wide temperature range, utilizing these findings to estimate the thermalization time of ultra-soft gluons during reheating and demonstrating significant deviations from perturbative axion estimates even at the electroweak scale.

The Gist

Imagine the universe right after the Big Bang as a chaotic, super-hot soup of energy. In this soup, there are tiny particles called gluons that act like the "glue" holding everything together. Sometimes, these gluons do something strange: they flip the universe's "magnetic switch." This flipping is called a sphaleron transition.

Think of a sphaleron transition like a hiker trying to cross a mountain pass.

• The Valley: A stable state of the universe.
• The Mountain: A barrier of energy separating two valleys.
• The Hiker: A gluon field.
• The Sphaleron: The exact peak of the mountain. If the hiker has enough energy to reach the peak, they can roll down into the other valley, changing the fundamental nature of the universe in that spot.

This paper is about measuring how often these hikers manage to cross the mountain in two different scenarios: when the soup is perfectly mixed and hot (thermal), and when it's a chaotic, messy explosion (non-thermal).

Here is a breakdown of their findings using simple analogies:

1. The Two Scenarios: The Calm Lake vs. The Tsunami

The researchers used supercomputers (lattices) to simulate the universe in two states:

• Scenario A: The Thermal Lake (Equilibrium)
  Imagine a calm lake where the water molecules are jiggling randomly but evenly. This represents the universe after it has had time to settle down. The researchers calculated how often the "hikers" (gluons) cross the mountain here. They found that at extremely high temperatures (trillions of degrees), their computer results matched the old math predictions. However, at lower (though still incredibly hot) temperatures, the "hikers" crossed the mountain much more often than the old math predicted. It turns out the "soft" ripples in the water (soft gluons) are doing more work than we thought.

• Scenario B: The Tsunami (Non-Equilibrium)
  Imagine a massive wave crashing onto the shore. The water is chaotic, piled up, and moving wildly. This represents the universe immediately after inflation (the rapid expansion of the early universe), before it had time to settle.

  • The Finding: In this chaotic state, the "hikers" cross the mountain much faster than in the calm lake. The chaos actually helps the transitions happen more frequently.

2. The "Thermalization" Race: Who Settles Down First?

One of the big questions in cosmology is: How long does it take for the universe to cool down and become a normal "soup" after the Big Bang?

The authors compared the speed of the "hikers" in the calm lake vs. the tsunami.

• The Analogy: Imagine a room full of people running wildly (the non-thermal state). They need to stop running and sit down in chairs (thermalize).
• The Result: The researchers found that the "ultra-soft" particles (the tiny, slow-moving ones) settle down and form a calm "thermal bath" very quickly.
• Why it matters: This calm bath then acts as a cushion for the "hard" particles (the fast, energetic ones) to crash into and slow down.
• The Conclusion: For this to work, the universe must have been incredibly hot (at least $10^{10}$ GeV) during its "reheating" phase. If it wasn't hot enough, the fast particles would never slow down, and the universe wouldn't look the way it does today.

3. The Axion Mystery: The Ghost Particle

The paper also looks at Axions. Think of axions as "ghost particles" that are a leading candidate for Dark Matter (the invisible stuff holding galaxies together).

• The Problem: We don't know how many axions were created in the early universe.
• The Old Math: Scientists used to guess the number of axions created based on simple equations (perturbation theory).
• The New Discovery: The researchers found that the "soft" gluons (the ripples in the soup) create axions much more efficiently than the old math predicted.
  • The Analogy: It's like trying to count how many bubbles form in a soda. The old math counted only the big bubbles. The new math shows that the tiny, invisible bubbles (caused by the non-perturbative interactions) are actually responsible for 75% of the fizz!
• The Impact: Because so many more axions are being made than we thought, they might contribute more to the "radiation budget" of the early universe. However, the researchers calculated that even with this boost, the number of axions is still low enough that it doesn't break our current understanding of the universe (specifically, the number of neutrino species).

Summary of the "Big Picture"

1. We were wrong about the speed: In the chaotic early universe, the "switch-flipping" events (sphalerons) happen much faster than in a calm universe.
2. The Universe cools fast: The "soft" particles settle down quickly, creating a thermal bath that helps the rest of the universe cool and stabilize. This suggests the universe was extremely hot right after the Big Bang.
3. More Dark Matter candidates: The "ghost particles" (axions) are being produced more efficiently by these chaotic interactions than we previously thought, though not enough to break our current cosmological models.

In short, the authors used a digital microscope to look at the very first moments of the universe and found that the "chaos" of the early days was actually a very efficient factory for creating the conditions we see today.

Technical Summary

1. Problem Statement

The paper addresses two critical issues in high-temperature Quantum Chromodynamics (QCD) and early universe cosmology:

• Sphaleron Transition Rates: Accurately determining the rate of topological transitions (sphalerons) in $SU(N)$ gauge theories ($N=2, 3$) across a vast temperature range ($0.6 - 10^{15}$ GeV). While perturbative estimates exist for weak couplings, non-perturbative effects from "soft" magnetic gluons become dominant at lower temperatures (higher couplings), where standard perturbation theory fails.
• Axion Production: Calculating the thermal production rate of QCD axions. Previous perturbative calculations (Hard Thermal Loop resummation) may underestimate this rate because they neglect non-perturbative contributions from soft gluons and sphaleron transitions, which are crucial for determining the relic axion abundance and the effective number of neutrino species ($N_{eff}$).
• Thermalization in Reheating: Understanding the thermalization time of ultra-soft gluons produced during the reheating epoch after inflation, specifically comparing thermal equilibrium states with non-thermal, over-occupied "glasma-like" initial states.

2. Methodology

The authors employ Lattice Gauge Theory techniques within an Effective Field Theory (EFT) framework to overcome the limitations of standard lattice simulations at high temperatures.

• Effective Theory for Soft Gluons:

  • Instead of simulating the full QCD theory, they utilize a 3D effective theory for soft magnetic gluons (momenta $\lesssim g^2T$).
  • This EFT integrates out hard ($|p| \sim \pi T$) and semi-hard ($|p| \sim gT$) modes, incorporating their effects via a color conductivity ($\sigma$) and stochastic noise ($\zeta$) in the Langevin equation of motion (Eq. 6).
  • This approach avoids ultraviolet divergences inherent in classical Hamiltonian simulations and allows for large spatial volumes necessary to capture diffusive sphaleron dynamics.

• Simulation Protocols:

  • Thermal Case: The system is evolved using a leap-frog integrator with stochastic noise to generate thermal gauge configurations. The Gauss law constraint is maintained with high precision ($10^{-15}$).
  • Non-Thermal Case: Simulations start from "glasma-like" initial conditions where infrared gluons are over-occupied ($f(p) \sim 1/\alpha_s$). These evolve via classical Hamiltonian dynamics (without noise) to a self-similar scaling regime.
  • Chern-Simons Number ($N_{CS}$) Measurement: To extract the sphaleron rate, the authors use calibrated cooling to remove UV fluctuations and project configurations to the nearest vacuum. The change in $N_{CS}$ is calculated between time steps $t_1$ and $t_2$ using improved electric and magnetic field definitions (O($a^2$) improved) to minimize discretization errors.

• Axion Production Calculation:

  • The production rate is derived from the two-point correlation function of the topological charge density ($G\tilde{G}$).
  • The rate is computed non-perturbatively on the lattice and compared against perturbative HTL (Hard Thermal Loop) estimates.

3. Key Contributions

• New Lattice Data: The first comprehensive lattice determination of sphaleron rates for $SU(2)$ and $SU(3)$ gauge theories spanning temperatures from $0.6$ GeV to $10^{15}$ GeV.
• Non-Thermal vs. Thermal Comparison: Novel results comparing sphaleron rates in thermal plasmas versus non-thermal, over-occupied plasmas with identical energy densities.
• Thermalization Timescale: A quantitative estimate of the time required for ultra-soft gluons to thermalize during the reheating epoch, derived by matching thermal and non-thermal sphaleron rates.
• Non-Perturbative Axion Rate: A demonstration that axion production is dominated by non-perturbative magnetic gluons even at the electroweak scale, significantly deviating from perturbative predictions.

4. Key Results

A. Sphaleron Rates ($\Gamma_{sph}$)

• Thermal Plasma: The lattice results agree with the parametric perturbative estimate ($\Gamma_{sph} \propto g^{10}T^4 \ln(1/g)$) only at very high temperatures ($T > 10^8$ GeV for $SU(2)$ and $T > 10^{10}$ GeV for $SU(3)$), where the coupling $g$ is sufficiently small ($g \lesssim 0.6$). At lower temperatures (higher $g$), non-perturbative effects cause significant deviations.
• Non-Thermal Plasma: In over-occupied non-thermal plasmas, the sphaleron rate follows a scaling law $\Gamma_{sph} \sim Q_s^4 (Q_s t)^{-1.2}$, where $Q_s$ is the saturation scale.
• Comparison: At asymptotically high temperatures (weak coupling), thermal rates are suppressed compared to non-thermal rates due to damping from hard gluons. However, at lower temperatures ($g \gtrsim 1$), the trend reverses, with thermal rates becoming higher.

B. Thermalization Time in Reheating

By equating the energy densities of thermal and non-thermal plasmas, the authors estimate the thermalization time ($t_{th}$) for ultra-soft gluons:

• Scaling: $t_{th} \sim g^{-7.2} T^{-1}$.
• Magnitude: For reheating temperatures $T \sim 10^9 - 10^{13}$ GeV, $t_{th}$ is extremely short ($1/t_{th} \sim 10^7 - 10^{11}$ GeV).
• Implication: Ultra-soft gluons thermalize much faster than hard gluons split into softer ones ($t_{th} \ll t_{hard}$) only if the reheating temperature $T_R \gtrsim 10^{10}$ GeV. If $T_R < 10^{10}$ GeV, the hierarchy breaks down, challenging standard perturbative reheating scenarios.

C. Axion Production

• Deviation from Perturbation Theory: The non-perturbative lattice calculation shows that axion production rates deviate significantly from HTL perturbative estimates even at the electroweak scale.
• Dominance of Sphalerons: For couplings $g \gtrsim 2/3$, the contribution from sphaleron transitions (magnetic gluons) accounts for nearly 75% of the total production rate.
• Relic Yield: Solving the Boltzmann equation with these non-perturbative rates, the authors find that axions remain in thermal equilibrium if $T_R \gtrsim 10^9$ GeV.
• Cosmological Constraints: The resulting contribution to the effective number of neutrinos is $\Delta N_{eff} \approx 0.027$, which is well within the Planck bound ($\Delta N_{eff} \leq 0.28$).

5. Significance

• Cosmological Reheating: The study provides a rigorous constraint on the reheating temperature of the early universe. It suggests that for perturbative reheating to be a valid description where soft modes thermalize quickly to form a bath for hard modes, $T_R$ must be at least $\sim 10^{10}$ GeV. This supports Higgs inflation scenarios.
• Axion Dark Matter: The results highlight that accurate predictions for axion relic abundance and $N_{eff}$ require non-perturbative lattice calculations, as perturbative methods fail to capture the dominant magnetic sector contributions at relevant energy scales.
• Methodological Advance: The work demonstrates the efficacy of 3D effective field theories on the lattice for studying real-time dynamics in high-temperature gauge theories, bridging the gap between perturbative QCD and classical statistical simulations.