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Higher topological charge in the QCD vacuum and axion cosmology

This paper demonstrates that gauge field configurations with higher topological charge modify the QCD vacuum's $\theta$ -dependence, leading to characteristic temperature variations in topological susceptibilities and a multi-instanton mechanism that enhances the production of axion dark matter.

Original authors: Fabian Rennecke

Published 2026-07-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Fabian Rennecke

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: The Quantum Vacuum is a Busy City

Imagine the vacuum of our universe (empty space) not as a silent, empty void, but as a bustling city. In the world of Quantum Chromodynamics (QCD)—the physics that holds atoms together—this city is filled with invisible, swirling energy fields.

Usually, physicists think of this city as having a very simple, repetitive layout. They imagine the "buildings" (called instantons) are all single-story houses. These houses represent tiny, fleeting events where the vacuum tunnels from one state to another.

The Paper's New Idea:
This paper suggests that the city is actually much more complex. Besides the single-story houses, there are also multi-story skyscrapers (called multi-instantons). These represent events where the vacuum tunnels through multiple layers at once. The author argues that even though these skyscrapers are rare, they change the fundamental "architecture" of the vacuum in ways that single-story houses cannot.

Part 1: The "Theta" Angle and the Compass

The vacuum has a hidden setting called $\theta$ (theta). You can think of $\theta$ as a compass needle pointing in a specific direction.

In the standard view, the vacuum only cares about the compass pointing North or South (a simple up-and-down wobble).
The author shows that when you include the "skyscrapers" (higher topological charges), the compass doesn't just wobble simply. It gets wobbly and distorted.

The Analogy:
Imagine a guitar string.

Single Instantons: The string vibrates at a single, pure note (a fundamental frequency).
Multi-Instantons: The string now vibrates with overtones (harmonics). The sound is richer, more complex, and slightly "out of tune" with the simple note.

The paper calculates how these overtones change the "sound" of the vacuum. This changes how the vacuum reacts to the compass needle ( $\theta$ ), making the relationship between the angle and the energy of the vacuum non-linear (anharmonic).

Part 2: The Axion – The Universe's "Relaxation" Mechanism

There is a famous mystery in physics: Why doesn't this compass needle ( $\theta$ ) cause a violation of symmetry (CP violation) that we should see in nature?

The Solution: Physicists proposed a new particle called the axion.
The Metaphor: Imagine the axion is a relaxing spring attached to the compass. If the compass is stuck at a weird angle, the spring pulls it toward zero (North). As the spring relaxes, it settles into a calm state. This process is called vacuum realignment.

The paper asks: Does the presence of the "skyscrapers" (multi-instantons) change how this spring behaves?

The Answer:
Yes. Because the skyscrapers distort the "landscape" of the vacuum (the potential energy), the spring's path changes.

Flattening the Hill: In some spots, the multi-instantons make the "hill" the spring has to roll down very flat.
The Freeze: If the spring starts on this flat spot, it gets stuck there for a long time. It doesn't roll down immediately.
The Result: Because it stays stuck longer, it builds up more energy before it finally starts rolling and oscillating.

The Cosmological Impact:
This extra energy means that if axions make up Dark Matter (the invisible stuff holding galaxies together), there might be more of it than we previously thought. The "skyscrapers" act as a mechanism that boosts the amount of axion dark matter in the universe.

Part 3: The "Toy Universe" Experiment

The author runs a simulation to prove this point.

The Setup: He creates a "toy universe" (a simplified model) where the rules of the vacuum are governed by these multi-instantons.
The Observation: He watches the axion field evolve over time.
- Without skyscrapers: The axion starts rolling down the hill quickly and settles into a rhythm.
- With skyscrapers: The axion gets "frozen" on the flat part of the hill for a longer time. When it finally starts moving, it vibrates faster and carries more energy.
The Conclusion: The presence of these complex vacuum structures increases the final amount of axion dark matter.

Important Caveats (What the Paper Doesn't Say)

The paper is very careful to note the limits of its findings:

It's a "Toy" Model: The calculations were done in a simplified version of the universe called "quenched QCD" (where certain particles, like light quarks, are ignored). In the real universe, these light particles might suppress the "skyscraper" effects, making them very small.
High Temperatures Only: This theory works best in the "deconfined phase" of the universe, which happened when the universe was extremely hot (like just after the Big Bang). We don't know if these effects work the same way in our cold, current universe.
No Clinical Uses: This is purely theoretical physics about the early universe and the nature of empty space. It has no direct application to medicine, engineering, or daily life.

Summary

This paper suggests that the empty space of our universe is more complex than we thought. It's not just made of simple, single events, but also rare, complex "multi-events." These complex events distort the energy landscape of the universe, which could act like a topological mechanism to increase the amount of axion dark matter we have today. It's like finding out that the foundation of a building has hidden, complex supports that change how much weight the building can hold.

Technical Summary: Higher Topological Charge in the QCD Vacuum and Axion Cosmology

Problem Statement
The vacuum structure of Quantum Chromodynamics (QCD) is intrinsically linked to topological gauge field configurations, characterized by an integer topological charge $Q$ . While the standard description of the QCD vacuum at high temperatures (in the deconfined phase) relies on a dilute gas of single-instantons ( $|Q|=1$ ), this approximation neglects configurations with higher topological charges ( $|Q| > 1$ ). Although multi-instantons are exponentially suppressed in the weak-coupling limit ( $\sim e^{-8\pi^2|Q|/g^2}$ ), they represent distinct self-dual field configurations that cannot be simply interpreted as superpositions of single-instantons. The paper investigates how including these higher topological charge configurations modifies the $\theta$ -dependence of the QCD vacuum, the resulting topological susceptibilities, and the cosmological production of axion dark matter via the vacuum realignment mechanism.

Methodology
The study employs a semi-classical analysis within the deconfined phase at high temperatures ( $T \gtrsim 2.5T_c$ ), where the topological structure is modeled as a dilute gas of instantons with arbitrary topological charge $Q$ .

Partition Function Derivation: The author derives the contribution of multi-instantons to the QCD partition function $Z_Q$ in the limit of Small Constituent Instantons (SCI). In this limit, the size parameters $\rho_i$ of the constituent instantons are small compared to their separations $|R_{ij}|$ .
- Using the ADHM construction, exact multi-instanton solutions are expanded systematically.
- The analysis distinguishes between "genuine" multi-instanton processes (where constituent instantons overlap) and processes that factorize into single-instanton contributions.
- The calculation focuses on the next-to-leading order in the SCI expansion ( $\zeta^3$ , where $\zeta \sim \rho/|R|$ ). At this order, the gauge field contribution factorizes, but the quark zero mode determinant does not. The genuine multi-instanton contribution $\Delta Z_Q$ arises from the overlap of quark zero modes associated with different constituent instantons.
Free Energy and Susceptibilities: The $\theta$ -dependent free energy density $\Delta F(\theta)$ is constructed from the sum over all topological charges. Topological susceptibilities $\chi_{2n}$ are defined as the $2n$ -th derivatives of the free energy with respect to $\theta$ at $\theta=0$ . The paper evaluates these numerically for quenched QCD (without dynamical quarks) to isolate multi-instanton effects, as light quarks strongly suppress instanton density.
Axion Cosmology: The derived free energy is identified with the axion effective potential $V(\bar{\theta}/f_a)$ . The classical equation of motion for the axion field in a radiation-dominated universe is solved numerically. The study compares the evolution of the axion field and its resulting energy density (dark matter abundance) when the potential includes multi-instanton corrections versus the standard single-instanton cosine potential.

Key Contributions and Results

Modification of $\theta$ -Dependence: The inclusion of multi-instantons introduces anharmonic terms to the free energy. While single-instantons yield a potential proportional to $(1-\cos\theta)$ , multi-instantons generate a series of overtones proportional to $(1-\cos(Q\theta))$ . This results in a potential that deviates significantly from the simple harmonic form, particularly for larger values of the dimensionless parameter $\bar{Z}_1$ (related to the single-instanton density).
Topological Susceptibilities: In the single-instanton approximation, all topological susceptibilities are identical up to a sign. The inclusion of higher charges lifts this degeneracy. The paper demonstrates that multi-instanton effects amplify the temperature dependence of the susceptibilities $\chi_{2n}$ , causing them to decrease more rapidly as temperature drops before asymptoting to the single-instanton behavior at very high $T$ .
Anharmonicity Coefficients: The coefficients $b_{2n}$ , which parametrize deviations from the quadratic $\theta$ -dependence, are shown to be temperature-dependent in the presence of multi-instantons, whereas they are constant in the single-instanton limit. This temperature dependence qualitatively matches trends observed in lattice QCD simulations above the critical temperature.
Axion Dark Matter Production: The modified axion potential exhibits a flattening near the maxima (and potentially local minima) due to anharmonicities.
- Overdamping: This flattening reduces the curvature of the potential ( $V''$ ), keeping the axion field in an overdamped regime for a longer duration. The axion remains "frozen" at its initial misalignment angle for a longer period.
- Oscillation Frequency: Once the axion begins to oscillate, the steeper potential near the minimum (induced by multi-instantons) leads to a higher oscillation frequency.
- Result: Both effects—the prolonged overdamping (increasing potential energy) and the faster oscillation (increasing kinetic energy)—lead to a higher energy density of cold axion dark matter today compared to the single-instanton prediction.

Significance and Limitations
The paper claims that higher topological charge effects provide a microscopic explanation for the specific temperature dependence of topological susceptibilities observed in lattice QCD. Furthermore, it proposes a topological mechanism to increase the predicted abundance of axion dark matter.

However, the author maintains a modest stance regarding the quantitative impact in realistic QCD:

Suppression by Quarks: In full QCD with light dynamical quarks, the instanton density is significantly suppressed. Consequently, the multi-instanton corrections to the leading single-instanton behavior are found to be negligible for most practical purposes, affecting only very high-order susceptibilities.
Approximations: The results rely on the SCI limit and the dilute gas approximation. The author notes that the SCI expansion is valid only far from constituent instanton centers and that higher-order corrections and interactions between instantons (which are neglected) could alter the quantitative results.
Scope: The analysis is restricted to the deconfined phase at high temperatures. The nature of topological configurations in the confined phase remains unsettled and is not addressed.

In summary, the work establishes a formal framework for including arbitrary topological charges in the semi-classical description of the QCD vacuum, demonstrating that while these effects are conceptually significant and qualitatively consistent with lattice data, their quantitative impact on axion cosmology in full QCD may be small under the current approximations.