Gauging Axionic Symmetries and Dark Matter: In memory of George Lazarides

Written in memory of George Lazarides, this paper revisits a joint study on the cosmology of gauged axions within anomalous $U(1)$ models, demonstrating how Stueckelberg fields and Wess-Zumino terms generate a physical axion-like dark matter candidate through a distinct misalignment mechanism that requires a large Stueckelberg scale.

The Gist

This paper is a tribute to the late physicist George Lazarides, written by his colleague Claudio Corianò. It uses a specific scientific model to explore how the universe might be filled with "Dark Matter," but it does so by telling a story about how different parts of physics fit together.

Here is the explanation in simple, everyday language, using analogies to make the concepts clear.

The Big Picture: A Tribute to a "Cosmic Detective"

Imagine George Lazarides as a detective who never looked at a clue in isolation. If he found a piece of evidence about a particle (like an axion), he immediately asked: "How does this fit into the history of the universe? Does it create monsters (like unstable walls) or ghosts (like unwanted relics)?"

This paper revisits a project the author and George worked on together. They asked a simple question: What happens if the "axion" (a famous candidate for Dark Matter) isn't just a free-floating particle, but is actually tied to a force of nature that has been "gauged" (given a specific rulebook)?

The Cast of Characters

1. The Axion (The Invisible Hero): In standard physics, the axion is like a ghostly, invisible particle that solves a mystery about why the universe doesn't behave strangely with magnetism (the "Strong CP problem"). It is also a top candidate for Dark Matter.
2. The Stueckelberg Field (The Chameleon): In this specific model, there is a field called "Stueckelberg." Think of this as a chameleon. At high energies (early universe), it is invisible because it hides inside a force carrier (a gauge boson). It's not a real particle yet; it's just part of the machinery.
3. The Higgs (The Transformer): The Higgs field is famous for giving mass to particles. In this story, the Higgs acts like a mixer. When the universe cools down, the Higgs mixes with the "chameleon" Stueckelberg field.
4. The Axi-Higgs (The Born Again): After the Higgs and the Stueckelberg field mix, a new, real particle is born. The authors call this the Axi-Higgs. It is the physical version of the axion in this specific model.

The Story of the Universe (The Timeline)

The paper argues that the history of this Axi-Higgs is very different from a standard axion. It goes through two distinct "awakenings":

Phase 1: The Electroweak Awakening (The "Almost" Moment)

• The Event: When the universe was young and hot, the Higgs field turned on (Electroweak Symmetry Breaking).
• The Result: The Axi-Higgs finally became a physical particle.
• The Analogy: Imagine a seed sprouting. It's a real plant now, but it's tiny.
• The Outcome: Because it appeared so early and the "scale" was small, this first awakening produced almost zero Dark Matter. It was like a drop of water in an ocean.

Phase 2: The QCD Awakening (The "Real" Moment)

• The Event: Much later, when the universe cooled further, the strong nuclear force (QCD) started to interact with this particle.
• The Result: This interaction gave the particle a "mass" and made it start oscillating (wiggling) like a pendulum.
• The Analogy: This is like the seed finally growing into a massive oak tree.
• The Outcome: This is where the Dark Matter comes from. However, there is a catch. For this tree to grow big enough to fill the universe with Dark Matter, the "Stueckelberg scale" (the energy level where the chameleon was hiding) must be huge.

The Main Conclusion: The "Goldilocks" Scale

The paper's mathematical analysis leads to a very specific conclusion:

• If the hidden energy scale is too low (like the energy of the Large Hadron Collider, in the "TeV" range), the resulting Dark Matter is negligible. It's too small to matter.
• For the Axi-Higgs to be a significant source of Dark Matter, the hidden energy scale must be massive—around 10 million billion (10^7) GeV.

The Metaphor:
Think of the Stueckelberg scale as the size of a dam holding back water.

• If the dam is small (low energy), the water (Dark Matter) trickles out and disappears.
• If the dam is gigantic (intermediate scale), the water rushes out and fills the valley, creating a lake (Dark Matter abundance).

Why This Matters (The "George" Lesson)

The author emphasizes that this isn't just about calculating numbers. It's about George Lazarides's philosophy: You cannot understand a particle without understanding the "gauge structure" (the rules) it lives in.

In standard models, you might just assume an axion exists. In this model, the axion is a byproduct of a complex dance between forces, anomalies, and symmetry breaking. The paper shows that:

1. The "rules" of the universe (gauge symmetries) dictate when a particle becomes real.
2. The history of the universe (cosmology) dictates how much of that particle exists today.

Summary

This paper is a memorial that says: "George taught us that particles and the history of the universe are inseparable." By studying a specific model where the axion is "gauged," they found that this particle can only be the Dark Matter we see today if the universe had a very specific, high-energy setup in its early days. If that setup wasn't right, the particle would be there, but it would be too faint to be the Dark Matter holding galaxies together.

Technical Summary

Based on the memorial paper by Claudio Corianò, here is a detailed technical summary of the work on Gauging Axionic Symmetries and Dark Matter, dedicated to the memory of George Lazarides.

1. Problem Statement

The paper addresses the cosmological viability of axion-like particles (ALPs) arising from anomalous $U(1)$ gauge symmetries in effective field theories derived from string compactifications (specifically orientifold vacua).

• The Core Issue: In standard Peccei-Quinn (PQ) models, the axion is a global symmetry breaking mode. However, in models with anomalous $U(1)$ factors (common in Type-I and orientifold string vacua), the symmetry is gauged. The associated Stueckelberg pseudoscalar ($b$) is initially "eaten" by the gauge boson to generate mass, raising the question: Can a physical, light axion-like state survive after symmetry breaking, and can it serve as a viable Dark Matter (DM) candidate?
• The Domain Wall Problem: A historical concern in axion cosmology is the formation of stable domain walls if the discrete symmetry left by QCD instantons is not properly embedded. George Lazarides previously showed (with Shafi) that embedding this discrete symmetry into a continuous gauge group resolves this. This paper extends that philosophy to gauged axions, asking how the global vacuum structure and gauge realization alter the cosmological history of the axion.

2. Methodology and Theoretical Framework

The authors analyze a specific effective field theory (EFT) where the Standard Model (SM) gauge group is extended by an anomalous abelian factor, $U(1)_B$.

• Stueckelberg Mechanism: The $U(1)_B$ gauge boson ($B_\mu$) acquires mass via the Stueckelberg mechanism involving a pseudoscalar field $b$:
  $$\mathcal{L}_{St} = \frac{1}{2} (\partial_\mu b - M B_\mu)^2$$
  Here, $b$ is not a physical particle in the high-energy phase; it is absorbed into the longitudinal component of $B_\mu$.
• Anomaly Cancellation: The gauge anomaly is cancelled locally via Wess-Zumino (WZ) counterterms and generalized Chern-Simons terms. These terms couple the Stueckelberg field $b$ to gauge field strengths (e.g., $b F \tilde{F}$), ensuring gauge invariance.
• Higgs-Stueckelberg Mixing: The model includes two Higgs doublets ($H_u, H_d$). After Electroweak Symmetry Breaking (EWSB), the CP-odd sector mixes the Stueckelberg field $b$ with the phases of the Higgs fields.
• Physical State Identification: The authors identify a single physical pseudoscalar state, termed the "axi-Higgs" ($\chi$), which emerges from this mixing. The other CP-odd modes are absorbed as Goldstone bosons by the massive $Z$ and $Z'$ bosons.

3. Key Contributions and Mechanisms

A. The Nature of the Axi-Higgs

Unlike a standard invisible axion where the shift symmetry is global, the axi-Higgs arises from a gauged symmetry. Its interactions are fixed by the anomaly-cancellation requirements (WZ terms), not by arbitrary global couplings. The physical field $\chi$ only becomes distinct after the mixing of the Higgs and Stueckelberg sectors.

B. Sequential Misalignment

The paper introduces a novel cosmological history characterized by two distinct misalignment events, differing from the single misalignment of standard PQ axions:

1. Electroweak Misalignment:

   • Occurs at the EWSB scale ($T \sim 100$ GeV).
   • The relevant decay constant is $\sigma_\chi \sim v$ (the electroweak scale).
   • Result: The relic abundance produced at this stage is negligible because the field is "frozen" and the scale is too low to generate significant DM density.

2. QCD Misalignment:

   • Occurs at the QCD phase transition ($T \sim 1$ GeV).
   • The mixed $U(1)_B - SU(3)_C$ anomaly generates a QCD potential for $\chi$.
   • Crucially, the effective decay constant for the QCD interaction is not $\sigma_\chi$, but is enhanced by the Stueckelberg mass scale $M$:
     $$f_{QCD}^\chi \sim \frac{M^2}{v}$$
   • The mass of the axi-Higgs is suppressed by this large scale:
     $$m_\chi \sim \frac{\Lambda_{QCD}^2}{f_{QCD}^\chi} \sim \frac{\Lambda_{QCD}^2 v}{M^2}$$

4. Results

• Relic Abundance Dependence: The contribution of the gauged axion to the Dark Matter relic density is highly sensitive to the Stueckelberg mass scale $M$.
  • TeV Scale ($M \sim \text{TeV}$): The effective decay constant $M^2/v$ is too small (comparable to the electroweak scale), resulting in a negligible DM contribution.
  • Intermediate Scale ($M \sim 10^7$ GeV): When $M$ is sufficiently large, the ratio $M^2/v$ becomes comparable to the standard invisible axion decay constant ($f_a \sim 10^{9} - 10^{12}$ GeV). In this regime, the QCD misalignment mechanism produces an appreciable relic abundance consistent with observed Dark Matter.
• Mass Range: For $M \sim 10^7$ GeV, the axi-Higgs mass is extremely light, consistent with the "fuzzy" or ultra-light axion regime, but its production mechanism is distinct from standard PQ axions.

5. Significance and Legacy

• Conceptual Shift: The paper reinforces George Lazarides's insight that axion physics cannot be separated from the gauge structure of the theory. The "axion" is not just a global symmetry breaking mode; its existence, mass, and cosmological role are dictated by how the symmetry is gauged and how anomalies are cancelled.
• Vacuum Structure: It highlights that the global identification of vacua (gauge equivalence vs. distinct global vacua) determines whether topological defects (like domain walls) form and how the field evolves cosmologically.
• New DM Paradigm: It proposes a specific class of Dark Matter candidates where the "axion" is a remnant of a Stueckelberg-Higgs mixing in an anomalous gauge theory. This connects string-inspired effective actions (orientifolds) directly to cosmological observables.
• Methodological Lesson: The work demonstrates that the cosmological history of a particle is determined by the full sequence of symmetry breaking (Stueckelberg phase $\to$ Electroweak phase $\to$ QCD phase), rather than just the low-energy effective potential.

In summary, the paper establishes that while gauged axions in TeV-scale models do not constitute Dark Matter, they can become viable DM candidates if the Stueckelberg scale is pushed to the intermediate scale ($10^7$ GeV), effectively acting as a "heavy" axion decay constant generator. This realization underscores the deep interplay between particle physics model building, anomaly cancellation, and early universe cosmology.