Specially Embedding a Composite Axion Model

This paper proposes a novel post-inflationary composite axion framework that resolves the cosmological domain wall problem by embedding the relevant gauge groups into a larger product structure, which sets the domain wall number to unity and utilizes UV instanton effects to destabilize domain walls while preserving the solution to the strong CP problem.

The Gist

The Big Picture: Solving a Cosmic "Traffic Jam"

Imagine the universe is a giant city. Physicists have a theory about a special, invisible particle called the axion that acts like a traffic cop, fixing a glitch in the laws of physics (the "Strong CP problem") that otherwise would make the universe behave very strangely.

However, there's a catch. In the most popular version of this theory (the "composite axion"), the axion is born from a new, hidden force that binds particles together. The problem is that when this happens, it creates a cosmic traffic jam called Domain Walls.

Think of these Domain Walls as invisible, infinite sheets of energy that stretch across the entire universe. In the old models, there were too many of these sheets (more than one "layer" of traffic). Because they are stable, they never disappear. If they existed, they would pile up so much energy that they would crush the universe, causing it to collapse long before we could exist. This is the "Domain Wall Problem."

The Authors' Solution: A "Special Embedding"

The authors of this paper propose a clever architectural fix. Instead of building the axion's home (the hidden force) as a standalone structure, they propose building it as a special room inside a much larger, more complex building.

Here is the analogy:

• The Old Way: Imagine trying to build a house (the axion model) in a field. The blueprints accidentally create a foundation that supports 5 different, equally stable versions of the house. This creates a mess where 5 different realities overlap, creating those dangerous, crushing walls.
• The New Way (Special Embedding): The authors say, "Let's build that house inside a massive, multi-story skyscraper (a larger gauge group)."

By embedding the small axion model inside this larger "skyscraper," the rules of the building change. Even though the small house looks like it has 5 versions from the inside, the structure of the skyscraper forces it to only have one true, stable version.

How It Works: The "Bias" and the "Small Instanton"

In the old models, the universe was stuck in a state of indecision, choosing between multiple stable options (the domain walls). The authors introduce a mechanism called Small Instanton Effects.

Think of this like a slight tilt in the floor of the skyscraper.

• Before the tilt, a ball (the universe) could sit happily in any of 5 different valleys (the domain walls).
• The "Small Instanton" effect acts like a tiny, invisible hand that tilts the floor just enough so that 4 of those valleys become shallow slopes, and only one valley remains deep and stable.

This "tilt" is called a bias term. It forces the universe to roll down into that single, safe valley. Because there is now only one choice, the dangerous "walls" between the choices become unstable and collapse immediately. The traffic jam clears, and the universe is safe.

The "Exotic Hadrons" Problem: The Unwanted Guests

The paper also addresses a second issue. The new hidden force creates strange, heavy particles called exotic hadrons.

• The Problem: These are like heavy, charged guests at a party who refuse to leave. They are so heavy and stable that they would stick around forever, clumping together with normal matter and ruining the chemistry of the universe (making it impossible for stars and life to form).
• The Fix: The authors suggest adding a "back door" (higher-dimensional operators). This allows these heavy guests to decay (leave the party) and turn into normal particles before they can cause any damage. They calculate that if the "back door" is open wide enough, these particles will vanish quickly enough to save the universe.

The Result: A Viable Universe

By using this "Special Embedding" technique:

1. The Strong CP Problem is solved: The axion still does its job of fixing the physics glitch.
2. The Domain Wall Problem is solved: The "tilt" ensures only one vacuum state exists, so the crushing walls never form.
3. Dark Matter: The axion can still exist as a candidate for Dark Matter (the invisible stuff holding galaxies together), though the authors note that in their specific setup, it might only make up part of the Dark Matter, not all of it.

Summary

The paper presents a new blueprint for the universe. It takes a promising but flawed theory (the composite axion) and hides it inside a larger, more complex mathematical structure. This structure naturally forces the universe to choose a single, safe path, eliminating the cosmic "traffic jams" (domain walls) that would have otherwise destroyed everything. It's a way of fixing a broken engine by installing it into a better-designed chassis.

Technical Summary

1. Problem Statement

The paper addresses two major cosmological challenges associated with post-inflationary composite axion models:

1. The Domain Wall Problem: In standard composite axion models, the Peccei-Quinn (PQ) symmetry breaking leads to a domain wall number $N_{DW} > 1$ (typically $N_{DW} = N$, where $N$ is the rank of the confining gauge group). This results in the formation of stable domain walls after the QCD phase transition. These walls form a network that dominates the energy density of the universe, leading to an overclosed universe, which contradicts observations.
2. The Exotic Hadron (CHAMP) Problem: The new confining dynamics produce heavy, stable, colored hadrons (baryons and mesons). If these particles survive until the QCD phase transition, they bind with Standard Model (SM) quarks to form Charged Massive Particles (CHAMPs). These are strongly constrained by astrophysical and cosmological observations and cannot constitute viable dark matter.

While previous solutions often relied on ad-hoc explicit PQ-breaking terms or secondary inflation, the authors seek a UV-complete, natural solution that preserves the axion's role in solving the Strong CP problem.

2. Methodology and Framework

The authors propose a novel framework based on Special Embedding of gauge groups.

A. Special Embedding Concept

Instead of a standard subgroup embedding (based on Dynkin diagram truncation), the authors embed the Infrared (IR) confining gauge group ($SU(N)$) and the QCD color group ($SU(3)_c$) into a larger Ultraviolet (UV) gauge group, specifically $SU(4N)$, in a non-trivial way.

• Mechanism: In a special embedding, the representation content and group-theoretic indices are reorganized such that the Dynkin index of the PQ-charged fermions in the UV theory differs from that in the IR theory.
• Result: The UV theory naturally possesses a domain wall number $N_{DW}^{UV} = 1$, while the naive IR description suggests $N_{DW}^{IR} = N$.

B. Model Construction

The authors construct an explicit UV completion with the gauge symmetry:
$$SU(4N) \times SU(3)^2 \times SU(N)^2 \times U(1)_X$$

• Matter Content: The model introduces Weyl fermions ($\Psi, \bar{\Psi}$) in the fundamental/anti-fundamental of $SU(4N)$ and complex scalar fields ($\Phi, \Phi', \Phi''$).
• Symmetry Breaking: The scalars acquire Vacuum Expectation Values (VEVs) at a cutoff scale $\Lambda_{cut}$, breaking the UV symmetry down to the original composite axion model's symmetry ($SU(N)_V \times SU(3)_c \times U(1)_Y$).
• Matching Conditions: The authors derive matching conditions for gauge couplings and $\theta$-angles between the UV and IR theories. Crucially, the $\theta$-angle matching involves factors of $N$ and $4$ derived from the special embedding, altering the effective anomaly coefficients.

C. Solving the Problems

1. Domain Wall Resolution:
   • In the IR, the axion potential is dominated by QCD effects, suggesting $N$ degenerate vacua.
   • However, the UV theory contains small instanton effects from the $SU(4N)$ gauge dynamics. These effects generate an effective mass term for the fermions, inducing a bias term ($V_{bias}$) in the axion potential.
   • This bias term lifts the vacuum degeneracy, effectively setting the physical domain wall number to $N_{DW} = 1$. The domain walls become unstable and decay rapidly, avoiding the overclosure problem.
2. Exotic Hadron Decay:
   • To address the CHAMP problem, the authors introduce higher-dimensional operators suppressed by a cutoff scale $\Lambda_{cutoff}$.
   • These operators couple the exotic mesons and baryons to SM fields (quarks and Higgs), allowing them to decay before they can form stable bound states or dominate the energy density.

3. Key Contributions

• Novel UV Completion: The paper provides the first explicit realization of a composite axion model where the domain wall problem is solved via special embedding into a larger gauge group, rather than by ad-hoc explicit breaking.
• Controlled Bias Term: The bias term is not arbitrary; it arises naturally from non-perturbative $SU(4N)$ instanton effects. The magnitude is calculable and can be tuned to be large enough to destabilize walls but small enough not to spoil the Strong CP solution.
• Cosmological Viability: The authors perform a full cosmological analysis, calculating the axion abundance and the shift in the effective $\bar{\theta}_{QCD}$ parameter.
• Alternative Realizations: The paper explores alternative setups (tri-fundamental representations and large representations) to achieve similar special embedding structures, demonstrating the robustness of the idea.

4. Results and Analysis

• Parameter Space: The authors identify a viable parameter region where:
  • The PQ breaking scale $f_{PQ} \gtrsim 10^{10}$ GeV.
  • The initial $\theta$-angle $\bar{\theta}_c \lesssim 10^{-1}$ (or smaller if spontaneous CP violation is invoked).
  • The small instanton effects are sufficient to collapse domain walls before they dominate the universe.
• Axion Abundance:
  • Because the small instanton bias term induces axion oscillations at a temperature $T_1$ significantly higher than the QCD scale ($T_{QCD}$), the resulting axion relic abundance is typically lower than in standard post-inflationary scenarios.
  • In the minimal setup, the axion constitutes a sub-component of Dark Matter. To account for 100% of DM, additional mechanisms (like spontaneous CP violation to reduce $\bar{\theta}_c$) are required.
• Strong CP Constraint: The shift in the axion potential minimum ($\Delta \bar{\theta}_c$) induced by the bias term is calculated. The model satisfies the neutron Electric Dipole Moment (nEDM) constraint ($\Delta \bar{\theta}_c \lesssim 10^{-10}$) within the viable parameter space.
• Exotic Hadrons: The analysis confirms that with appropriate higher-dimensional operators, exotic hadrons decay rapidly, evading CHAMP constraints.

5. Significance

This work represents a significant advancement in axion physics by:

1. Resolving the Domain Wall Crisis: It offers a theoretically motivated, UV-complete solution to the most severe cosmological obstacle for post-inflationary composite axions.
2. Unifying Concepts: It connects the concepts of Grand Unification (via special embedding) with composite axion models, suggesting that the group-theoretic structure of the UV theory can simultaneously control the domain wall number and generate necessary bias terms.
3. Future Directions: The paper highlights that while the domain wall problem is solved, the axion quality problem (protection against Planck-suppressed PQ-violating operators) remains open. It suggests that special embedding could be a powerful tool to address both domain walls and axion quality simultaneously in future models.

In summary, the paper demonstrates that by embedding the composite axion sector into a larger $SU(4N)$ gauge structure, the cosmological dangers of domain walls can be naturally eliminated, rendering the post-inflationary composite axion a viable candidate for Dark Matter and a robust solution to the Strong CP problem.