From strong interactions to Dark Matter: the non-perturbative QCD sphaleron rate

This plenary talk, accepting the 2025 Kenneth G. Wilson Award, summarizes significant contributions to understanding topology in QCD and related gauge theories through algorithmic advancements to mitigate topological freezing, analyses of Dirac spectral properties, and investigations into axion phenomenology.

The Gist

The Big Picture: From Tiny Particles to the Dark Universe

Imagine the universe is a giant, chaotic kitchen. Inside this kitchen, there are tiny, invisible ingredients called quarks and gluons that make up protons and neutrons (the stuff in your body). These ingredients are glued together by a force called the Strong Interaction.

For decades, scientists have known that these ingredients can sometimes do something weird: they can "flip" their internal identity in a way that changes the fundamental structure of space itself. This paper is about measuring exactly how often this flipping happens, and why that number matters for two very different things:

1. Colliders: Exploding atoms in giant machines to see what happens.
2. Dark Matter: Trying to find the invisible stuff that holds galaxies together.

The author, Claudio Bonanno, won a prestigious award for figuring out how to measure this "flipping rate" using a supercomputer, something that was previously thought to be nearly impossible.

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1. The "Slippery" Problem: What is a Sphaleron?

To understand the paper, you first need to understand the concept of a Sphaleron.

Imagine a landscape of hills and valleys.

• The Valleys: These are stable states where the universe likes to sit. Let's call them "Valley A" and "Valley B."
• The Hill: Between them is a tall mountain.

Usually, if you are in Valley A, you stay there. To get to Valley B, you have to tunnel through the mountain (which is very hard and rare).

But, if you heat the universe up (like in the early days of the Big Bang or inside a particle collider), the "ground" starts shaking. Suddenly, the particles have enough energy to climb over the mountain instead of tunneling through it.

The Sphaleron is the very top of that mountain. It's an unstable, "slippery" spot (the name comes from a Greek word meaning "slippery" or "ready to fall"). Once a particle configuration reaches this peak, it can slide down into a new valley, changing the universe's properties.

The Big Question: How fast does this sliding happen? This speed is called the Sphaleron Rate.

2. Why Do We Care? (The Two Main Jobs)

The paper explains that knowing this "sliding speed" is crucial for two reasons:

Job A: The Chiral Magnetic Effect (The "Spin" Traffic Jam)

Imagine a highway where cars (particles) have a specific spin. If a "slippery" event happens, it creates an imbalance: suddenly, there are more cars spinning left than right.
If you then turn on a giant magnet (like the ones in heavy-ion collision experiments), all those left-spinning cars get pushed in one direction, and right-spinning cars in the other. This creates a massive electric current.

• The Analogy: It's like a traffic jam where the cars spontaneously organize themselves into lanes based on their spin, creating a flow of electricity.
• Why we need the paper: To predict how strong this current is, scientists need to know exactly how often the "slippery" events happen.

Job B: The Dark Matter Detective (The Axion)

There is a hypothetical particle called the Axion that is a leading candidate for Dark Matter.

• The Analogy: Imagine the universe is a giant bathtub filling up with water (Dark Matter). The water is being poured in by the "slippery" events (Sphalerons).
• Why we need the paper: To know how much Dark Matter exists today, we need to know how fast the faucet was turned on in the early universe. If we get the "slippery rate" wrong, our estimate of how much Dark Matter is in the universe will be wrong.

3. The Challenge: The "Reverse Movie" Problem

Here is the tricky part. We cannot watch these events happen in real-time on a computer.

• The Problem: Our supercomputers work in "Euclidean time" (a mathematical trick that makes calculations possible), but the Sphaleron rate happens in "Real Time" (Minkowski time).
• The Analogy: Imagine you have a movie of a car crash, but it's been scrambled into a pile of still photos. You need to reconstruct the speed of the crash just by looking at the photos.
• The Math: This is called an Inverse Problem. It is notoriously difficult because if your photos have even a tiny bit of blur (noise), your reconstruction of the speed could be completely wrong. It's like trying to guess the exact recipe of a soup just by tasting a spoonful that has a little bit of salt spilled on it.

4. The Solution: The "Smart Filter" (The HLT Method)

Claudio and his team developed a new mathematical tool (called the HLT method) to solve this "Reverse Movie" problem.

• The Old Way: Scientists used to guess the answer and hope the noise didn't ruin it.
• The New Way: They created a "Smart Filter." They tested thousands of different ways to reconstruct the speed, looking for a "Goldilocks Zone."
  • If the filter is too strict, it ignores the data.
  • If the filter is too loose, the noise ruins the answer.
  • They found the perfect balance where the answer stabilizes (a "plateau"). This is where they found the true Sphaleron rate.

They also had to deal with "Topological Freezing."

• The Analogy: Imagine trying to shuffle a deck of cards, but the cards are frozen together in a block. You can't change the order. As the computer simulations get more precise (finer resolution), the "cards" (the topology of space) get stuck, and the computer can't explore new possibilities.
• The Fix: They used a technique called Parallel Tempering. Imagine heating the deck of cards slightly to melt the ice, shuffling them, and then cooling them back down. This allowed them to explore the "landscape" effectively without getting stuck.

5. The Results: What Did They Find?

Using these new tools, they calculated the Sphaleron rate for the first time with full accuracy (including the effects of real-world quark masses).

• The Temperature Dependence: They found that as the universe gets hotter, the "slippery" events happen faster, but not in a simple straight line. It follows a specific curve that depends on the strength of the strong force.
• The Surprise: They found that adding "light" quarks (the lighter ingredients in the soup) didn't stop the slipping, contrary to what some simple theories suggested. The rate remained robust.

6. Why This Matters for the Future

This paper is a "Rosetta Stone" for future discoveries.

1. For Dark Matter: Now that we have a precise number for the "slippery rate," cosmologists can calculate exactly how much Axion Dark Matter should exist. This helps experimentalists know exactly what to look for.
2. For Colliders: It helps physicists understand the "fireballs" created in particle collisions, specifically how the Chiral Magnetic Effect works.
3. For Computing: It proves that we can solve these incredibly hard "Inverse Problems" using new algorithms, opening the door to solving other impossible physics puzzles.

Summary in One Sentence

Claudio Bonanno developed a new mathematical "smart filter" to reconstruct the speed of rare, slippery quantum events from scrambled computer data, giving us the precise numbers needed to understand how the universe created Dark Matter and how particle collisions work.

Technical Summary

1. Problem Statement

The paper addresses the calculation of the strong sphaleron rate ($\Gamma_S$) in Quantum Chromodynamics (QCD). This quantity represents the rate of real-time (Minkowski) transitions between topologically distinct vacua in the strong interaction sector.

• Physical Significance:
  • Collider Physics: $\Gamma_S$ governs the relaxation of chiral imbalances ($N_R - N_L \neq 0$) in hot, dense media (e.g., heavy-ion collisions). This is crucial for understanding the Chiral Magnetic Effect (CME).
  • Dark Matter (Axion Cosmology): The rate determines the production of hot axions in the early universe via the axion-gluon interaction. Accurate knowledge of $\Gamma_S$ is essential for predicting the relic abundance of axions, a leading Dark Matter candidate.
• Theoretical Challenge:
  • Lattice QCD simulations are formulated in Euclidean time, whereas $\Gamma_S$ is a real-time observable.
  • The relationship between the Euclidean topological charge correlator $G_E(\tau)$ and the spectral density $\rho(\omega)$ (from which $\Gamma_S$ is derived) is an ill-posed inverse problem.
  • Extracting $\Gamma_S$ requires inverting an integral equation (Eq. 9) where small statistical fluctuations in the lattice data can lead to massive errors in the output spectral density.
  • Previous attempts were limited to pure-gauge theories (no dynamical quarks) and lacked precision due to these numerical instabilities.

2. Methodology

The author and collaborators developed a novel approach combining advanced algorithmic techniques to solve the inverse problem and overcome sampling issues.

A. Solving the Inverse Problem: The HLT Method

Instead of traditional Maximum Entropy methods, the paper employs the Hansen–Lupo–Tantalo (HLT) method, a modification of the Backus–Gilbert approach.

• Formulation: The spectral density $\rho_L(\omega)/\omega$ is approximated as a linear combination of the lattice correlator $G_L(\tau)$ with unknown coefficients $g_\tau$.
• Optimization: The coefficients are determined by minimizing a functional $F_\lambda[g] = (1-\lambda)A^2[g] + \lambda B[g]$.
  • $A^2[g]$ measures the deviation of the "resolution function" (smearing kernel) from a target delta function.
  • $B[g]$ acts as a regulator, proportional to the statistical covariance of the lattice data, preventing the explosion of fluctuations.
• Strategy: By varying the parameter $\lambda$, the authors search for a "plateau" where the systematic error (smearing width) is controlled, and statistical errors have not yet diverged. This allows for a robust extraction of $\Gamma_S$ without relying on uncontrolled model assumptions.

B. Handling Topological Charge and Smoothing

• Smoothing: To remove UV divergences and renormalization issues inherent in lattice topological charge definitions, gauge configurations are smoothed (using cooling).
• Continuum Extrapolation: The authors perform a double extrapolation:
  1. Continuum limit ($a \to 0$): Removing lattice spacing artifacts.
  2. Smoothing radius limit ($R_s \to 0$): Ensuring the physical result is independent of the artificial smoothing scale.
• Sampling: At high temperatures ($T > T_c$), topological charge fluctuations are exponentially suppressed, leading to "topological freezing." The authors utilize a multicanonical algorithm to enhance the sampling of topological sectors, ensuring ergodicity.

C. Future Outlook: Parallel Tempering on Boundary Conditions (PTBC)

To reach higher temperatures (GeV scale) and finer lattice spacings required for axion cosmology, the paper introduces Parallel Tempering on Boundary Conditions (PTBC). This algorithm swaps configurations between simulations with different boundary conditions to drastically reduce autocorrelation times, effectively solving the topological freezing problem on very fine lattices ($a \sim 0.01$ fm).

3. Key Contributions

1. First Non-Perturbative Calculation in Full QCD: The paper presents the first reliable determination of $\Gamma_S$ in $N_f = 2+1$ QCD with physical quark masses (pions $\approx 135$ MeV) in the temperature range $1.5 \lesssim T/T_c \lesssim 4$.
2. Methodological Breakthrough: Successful application of the HLT method to the sphaleron rate, demonstrating that the ill-posed inverse problem can be solved with controlled systematic errors.
3. Cross-Validation: The method was cross-checked against pure-gauge results, showing perfect agreement with previous studies that used different methods, validating the new approach.
4. Algorithmic Innovation: Demonstration of the PTBC algorithm's superiority over standard RHMC (Hybrid Monte Carlo) in overcoming topological freezing, enabling simulations at finer lattice spacings.

4. Key Results

• Temperature Dependence: The sphaleron rate $\Gamma_S$ was calculated at 5 temperatures. The results show that, unlike the topological susceptibility, $\Gamma_S$ is not suppressed by the presence of light dynamical quarks.
• Scaling Behavior:
  • The data was fitted to two ansätze:
    1. Semiclassical prediction: $\Gamma_S/T^4 \propto [\log(T/T_c)]^{-C}$. Fixing $C=5$ (semiclassical) fits well, but allowing $C$ to float yields $C \approx 5.6$ with large uncertainty.
    2. Power-law fit: $\Gamma_S/T^4 \propto (T/T_c)^C$. This yields $C \approx 2.19(38)$.
  • The authors conclude that the current temperature range ($T \lesssim 4 T_c$) is insufficient to definitively distinguish between these functional forms; calculations at $T \sim 1$ GeV are needed.
• Systematic Control: The results exhibit clear plateaus with respect to the smoothing radius ($R_s$) and the smearing width ($\sigma$), confirming that the extracted values are robust and free from dominant systematic biases.

5. Significance

• For Phenomenology: This work provides the first ab initio input for calculating the axion production rate in the early universe. This is critical for narrowing down the parameter space for axion Dark Matter searches.
• For Heavy-Ion Physics: The results offer theoretical constraints for the Chiral Magnetic Effect in heavy-ion collisions, aiding the interpretation of experimental data from RHIC and LHC.
• For Lattice Field Theory: The paper establishes a new standard for computing real-time observables from Euclidean lattice data. The combination of the HLT method, multicanonical sampling, and PTBC algorithms provides a roadmap for tackling other difficult non-perturbative problems involving topology and real-time dynamics.

In summary, this paper bridges the gap between fundamental QCD topology and cosmological Dark Matter models by providing the first high-precision, non-perturbative calculation of the strong sphaleron rate in physical QCD, supported by novel algorithmic solutions to long-standing numerical challenges.