⚛️ high-energy theory

Revisiting wormhole-induced global symmetry breaking

This paper argues that the conclusion that wormholes explicitly break global symmetries is premature, demonstrating that the ensemble average over symmetry-breaking parameters is typically dominated by the symmetric point due to doubly exponential suppression of other configurations, thereby resolving the conventional axion quality problem in standard Peccei–Quinn models.

Original authors: Kiyoharu Kawana

Published 2026-07-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Kiyoharu Kawana

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

Imagine the universe as a giant, cosmic game of "connect the dots," but instead of paper, the dots are different versions of reality. For a long time, physicists believed that in the ultimate rulebook of quantum gravity, there are no "global symmetries." Think of a global symmetry like a perfect, unbreakable rule that says, "No matter where you are in the universe, this specific dance move must always be the same." The old theory suggested that invisible, tunnel-like shortcuts through spacetime called wormholes act like mischievous pranksters. They slip through these tunnels, steal the "dance moves" from one universe, and drop them in another, effectively breaking the rule and ruining the symmetry.

This prankster behavior was thought to create a specific set of "knobs" or dials in the universe's settings, called $\alpha$ -parameters. If these dials were turned to any number other than zero, the symmetry would be broken, and the universe would be a messy, rule-breaking place. This was particularly bad news for a particle called the axion (a candidate for dark matter), because if the symmetry protecting it was broken, the axion would behave in a way that contradicts what we see in experiments. This is known as the "axion quality problem."

The Big Twist: The Cosmic Vote

In this paper, Kiyoharu Kawana suggests that we've been looking at the pranksters all wrong. Just because the wormholes can turn the dials to random numbers doesn't mean they do.

Think of the $\alpha$ -parameters not as fixed settings, but as a massive, cosmic voting booth. Every possible setting of the dials casts a vote. The paper argues that when you count all the votes (an "ensemble average"), the result isn't a chaotic mix. Instead, the vote is overwhelmingly dominated by a single, specific outcome: the dials stay at zero.

Why? The paper uses a concept called the "fine-tuning mechanism." Imagine a landscape of hills and valleys representing the energy of the universe. The system naturally wants to roll down to the lowest valley (the most stable state). The paper shows that for many types of matter (specifically those where the symmetry is already spontaneously broken, like in standard models), the deepest, most stable valley happens to be exactly where the dials are at zero.

The "Double-Exponential" Silence

What about the other dials? What if the wormholes try to turn them to non-zero numbers? The paper suggests that the universe effectively "silences" these options. The probability of the universe settling into a state where the dials are turned away from zero is suppressed by a factor so incredibly tiny it's hard to even imagine: $\exp(-e^{2S_{ins}})$ .

To put that in perspective, if $e^{2S_{ins}}$ is a large number, this factor is like trying to find a single specific grain of sand on a beach, but then having to find that exact grain inside a beach that is itself made of grains of sand, repeated infinitely. It's a "doubly exponential" suppression. The paper argues that in the thermodynamic limit (when the universe is huge), the contributions from these broken-symmetry states are so negligible they might as well not exist.

The Verdict on the Axion

So, what does this mean for the axion? The paper concludes that in standard models where the symmetry is spontaneously broken, the "vote" of the universe lands squarely on the symmetric point ( $\vec{\alpha} = 0$ ).

This suggests that the axion quality problem does not arise in this wormhole-induced theory. The wormholes don't break the symmetry because the universe's own energy structure forces the symmetry-breaking dials to stay at zero. The "pranksters" are effectively neutralized by the sheer weight of the universe's preference for stability.

What the Paper Does Not Say

It is important to note what this paper does not claim. It does not say that wormholes don't exist, nor does it say that global symmetries are always safe. The paper explicitly argues against the idea that the mere existence of the $\alpha$ -parameters means the symmetry is broken. Instead, it suggests that the outcome depends on the specific phase structure of the matter in the universe.

The paper also doesn't claim to have measured this in a lab or simulated it on a supercomputer. Instead, it uses mathematical arguments and "toy models" (simplified versions of the universe) to show that under broad, reasonable situations, the symmetric point dominates. It suggests that the conventional view—that wormholes automatically break symmetries—is premature.

The Baby Universe Connection

The paper also touches on a concept called the "baby universe hypothesis," which suggests that tiny, disconnected universes (baby universes) might be popping in and out of existence. If there were many different baby universes, they could carry different values for these dials, leading to a loss of information. However, the paper suggests that even if there are many baby universes, the "fine-tuning" mechanism effectively selects just one specific state (the one where the dials are zero). In this way, the universe avoids the information loss problem, behaving as if there is only one baby universe, even if there are many.

In Summary

The paper proposes a new way to look at the universe's rules. Instead of wormholes being the agents that break global symmetries, the universe's own energy landscape acts like a magnet, pulling the symmetry-breaking parameters back to zero. This suggests that the axion might be safe after all, and the "axion quality problem" might be a non-issue in the wormhole-induced effective theory. The authors suggest that the fate of global symmetry isn't determined by the wormholes themselves, but by which "critical point" (the most stable state) wins the cosmic vote.

Technical Summary: Revisiting Wormhole-Induced Global Symmetry Breaking

Problem Statement
It is a widely held conjecture in quantum gravity that exact global symmetries cannot exist in a consistent theory. A primary mechanism proposed to enforce this is the contribution of Euclidean wormholes (or gravitational instantons) to the path integral. These wormholes generate bilocal terms in the effective action, which can be converted into local symmetry-breaking operators via auxiliary $\alpha$ -parameters. Conventionally, the mere appearance of these $\alpha$ -parameters is interpreted as evidence that wormholes explicitly break global symmetries. This leads to significant theoretical challenges, most notably the "axion quality problem," where wormhole-induced potentials must be suppressed sufficiently to avoid violating experimental bounds on the QCD axion's CP-violating phase.

Methodology
The paper revisits this mechanism by shifting the focus from the existence of $\alpha$ -parameters to the ensemble average over these parameters. The authors formulate the problem within a general matter theory possessing a $U(1)$ $p$ -form global symmetry coupled to gravity.

Effective Theory Formulation: The authors start with the Euclidean partition function including wormhole solutions. Summing over wormholes with different charges generates bilocal terms, which are localized using Gaussian integrals over complex $\alpha$ -parameters. This results in a Lorentzian partition function $Z_M$ (interpreted as a micro-canonical or multi-verse ensemble) where the $\alpha$ -parameters act as coupling constants for symmetry-breaking operators.
Thermodynamic Limit Analysis: The core of the analysis involves evaluating the integral over the $\alpha$ -parameters in the thermodynamic limit ( $V \to \infty$ ). The authors employ a "fine-tuning mechanism" analogous to the equivalence of ensembles in statistical mechanics. They demonstrate that the integral is sharply dominated by critical points of the vacuum energy density $\rho(\vec{\alpha})$ .
Critical Point Classification: The authors analyze the behavior of $\rho(\vec{\alpha})$ $ρ (α)$ based on the phase structure of the underlying matter sector:
- Saddle Points: Where $\partial \rho / \partial \alpha = 0$ .
- Phase Transition Points: Where the first derivative of $\rho$ is discontinuous (first-order transitions) or higher derivatives are discontinuous.
Suppression Mechanism: The authors compare contributions from the dominant critical point against other potential critical points (where $\vec{\alpha} \neq 0$ ). They show that non-dominant points are suppressed by a doubly exponential factor, $\exp(-e^{2S_{ins}})$ , where $S_{ins}$ is the instanton action, rather than the standard single exponential suppression.

Key Contributions and Results

Ensemble Dominance: The paper establishes that the fate of a global symmetry is not determined by the mere existence of symmetry-breaking operators but by which critical point in the $\vec{\alpha}$ -space dominates the ensemble average.
Symmetry Preservation in Spontaneously Broken Phases: The authors identify two distinct outcomes based on the original matter sector:
1. If the $U(1)[p]$ symmetry is not spontaneously broken in the original theory, the dominant critical point may be at $\vec{\alpha} = 0$ (preserving symmetry) or $\vec{\alpha} \neq 0$ (breaking symmetry), depending on the specific potential shape.
2. Crucially, if the $U(1)[p]$ symmetry is spontaneously broken in the original matter sector, the dominant critical point is always at $\vec{\alpha} = 0$ . In this scenario, the ensemble average dynamically localizes the couplings to zero, effectively restoring the global symmetry in the effective theory despite the presence of wormhole-induced operators.
Resolution of the Axion Quality Problem: Applying the above result to standard Peccei-Quinn (PQ) models (where $p=0$ and the PQ symmetry is spontaneously broken), the authors conclude that the symmetric point $\vec{\alpha} = 0$ dominates the ensemble. Consequently, the wormhole-induced symmetry-breaking terms are suppressed by the doubly exponential factor, rendering them negligible. This implies that the conventional axion quality problem does not arise in the wormhole-induced effective theory for standard PQ models.
Baby Universe Hypothesis: The results support the baby universe hypothesis (that the Hilbert space of baby universes is one-dimensional) from a dynamical perspective. Even if the Hilbert space dimension is greater than one, the thermodynamic limit causes the system to effectively select a single $\alpha$ -state (the critical eigenvalue), preventing information loss associated with superpositions of different baby universe sectors.
Connection to Multi-Critical Point Principle (MPP): The analysis aligns with the MPP, suggesting that symmetry-enhanced points often emerge as the dominant critical points in generalized ensembles, potentially resolving issues like quadratic divergences.

Significance
The paper argues that the conclusion that wormholes explicitly break global symmetries is premature. By rigorously analyzing the ensemble average, the authors demonstrate that the vacuum structure of the matter sector dictates the symmetry fate. Specifically, for theories with spontaneous symmetry breaking (like standard axion models), the wormhole mechanism does not necessarily lead to explicit symmetry breaking. This provides a natural, non-ad-hoc solution to the axion quality problem, removing the need for extreme fine-tuning of instanton actions or non-minimal extensions of the Standard Model to satisfy experimental constraints. The work refines the conventional interpretation of wormhole effects, shifting the focus from the presence of breaking operators to the statistical dominance of the symmetric vacuum in the $\alpha$ -ensemble.

Technical Summary: Revisiting Wormhole-Induced Global Symmetry Breaking

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