The structure of multi-axion solutions to the strong CP problem

This paper investigates the general structure of multi-axion solutions to the strong CP problem, deriving a universal sum rule that reveals diverse phenomenological patterns and mass-coupling relations beyond the standard single-axion paradigm, thereby motivating an expanded experimental search program.

The Gist

The Big Mystery: The "Silent" Neutron

Imagine the universe is a giant, complex machine. One of its most important parts is the Strong Force, which holds the nucleus of atoms together. Physicists have a rulebook (the Standard Model) that predicts this force should sometimes break a fundamental symmetry called CP symmetry. If this symmetry breaks, it's like a clock running backward; it should cause a tiny particle called a neutron to act like a tiny magnet with a north and south pole (an electric dipole moment).

However, when scientists look at neutrons, they are perfectly balanced. They don't act like magnets. The "clock" isn't running backward. This is the Strong CP Problem: Why is the universe so perfectly balanced when the rules say it shouldn't be?

The Usual Hero: The Single Axion

For decades, the favorite solution to this mystery has been a particle called the Axion. Think of the Axion as a "cosmic thermostat."

• How it works: Imagine the Strong Force has a dial that is stuck in the "wrong" position. The Axion is a field that naturally slides this dial until it hits the "zero" position, fixing the problem.
• The Prediction: In the standard story, there is only one Axion. Its mass (how heavy it is) and how strongly it talks to light (photons) are locked together in a specific way. Scientists call this the "QCD Band." It's like a narrow highway on a map where the Axion must be driving.

The New Idea: A Traffic Jam of Axions

This paper argues that we might be looking at the wrong map. Instead of a single Axion driving down a single highway, there might be a whole fleet of Axions (multi-axion systems).

The authors suggest that in many theories (especially those trying to explain the universe at its smallest scales, like String Theory), nature doesn't just give us one Axion. It gives us several. When you have a fleet, the rules change:

1. They can split up: The Axions can interact with each other, mixing their properties.
2. They can go off-road: Some Axions might end up heavier or lighter than the standard prediction.
3. They can drive anywhere: Some might end up to the right of the highway (heavier/weaker coupling) and some to the left (lighter/stronger coupling).

The "Sum Rule": The Cosmic Budget

The paper introduces a new mathematical tool called a "Sum Rule."

Imagine the Axion fleet has a shared budget.

• In the old "single Axion" theory, the budget is strict: The total "mass-coupling" of the Axion must equal exactly 1.
• In this new "multi-Axion" theory, the budget is more flexible.
  • If the Axions are all very heavy and far from the standard highway, the total budget might be less than 1.
  • If the Axions are light and interact strangely with light, the total budget might be more than 1.

This sum rule is a diagnostic tool. If scientists find multiple Axions and add up their properties, the result tells them how the universe was built at the very beginning (the "UV completion").

The Two Main Scenarios

The paper identifies two main ways these Axion fleets can behave:

1. The "Mirror World" Scenario (General-PQ Systems)
Imagine a mirror image of our universe exists alongside ours. The Strong Force in this mirror world also has a "wrong dial."

• The Result: The two worlds interact. The "thermostat" (Axion) has to fix both dials at once.
• The Effect: This can push the Axion to the left of the standard highway. It becomes very light and might be easier to find with current experiments. It's like a hidden Axion that is actually more visible than the standard one.

2. The "Big Strong Sector" Scenario
Imagine the Strong Force isn't just one force, but a combination of two or more larger forces that merged together (like two rivers joining).

• The Result: The Axion has to navigate this complex merging.
• The Effect: This often pushes the Axions to the right of the highway. They become heavier and harder to detect, requiring new, more sensitive experiments.

Why This Matters for Experiments

Currently, most experiments are built to hunt for the "Single Axion" driving down the narrow highway.

• The Paper's Warning: If we only look at the highway, we might miss the whole fleet.
• The Opportunity:
  • If we find Axions to the left (lighter), it suggests the universe has "mirror" partners or complex internal structures.
  • If we find Axions to the right (heavier), it suggests the Strong Force emerged from a larger, more complex system.
  • If we find Axions with different "anomaly ratios" (a fancy way of saying they talk to light differently than expected), it could prove that the universe's forces (like electromagnetism and the strong force) were never unified into a single simple group.

The Bottom Line

This paper is a blueprint for a broader search. It tells experimentalists: "Don't just look for the one Axion on the highway. Look everywhere. The universe might be driving a whole convoy of them, and finding them will tell us exactly how the fundamental laws of physics were written."

It doesn't promise a cure for a disease or a new engine; it promises a new way to read the "source code" of the universe by looking for a fleet of particles instead of just one.

Technical Summary

Technical Summary: The Structure of Multi-Axion Solutions to the Strong CP Problem

Problem Statement
The absence of CP violation in Quantum Chromodynamics (QCD) remains a significant fine-tuning puzzle in the Standard Model (SM), known as the Strong CP problem. The canonical solution involves the Peccei-Quinn (PQ) mechanism, which introduces a single axion field. This axion acquires a mass and a photon coupling determined by non-perturbative QCD effects and the ratio of electromagnetic to QCD anomaly coefficients ($E/N$). These relations define a specific "QCD band" in the axion mass-photon coupling ($m_a - g_{a\gamma}$) parameter space, which is the primary target of current experimental searches. However, theoretical extensions of the SM, including string theory, extra dimensions, and models with additional confining sectors, naturally predict multi-axion systems. Existing analyses of these systems often rely on restrictive assumptions, such as the existence of a PQ symmetry broken only by QCD (QCD-PQ systems) and universal anomaly coefficients ($E/N$) across all axions. These assumptions may fail to capture the full phenomenological landscape, potentially missing viable solutions to the Strong CP problem that lie outside the canonical QCD band.

Methodology
The authors develop a general analytical framework for $N$-axion systems where multiple $U(1)_{PQ}$ symmetries are anomalous under QCD. The methodology proceeds as follows:

1. General Parameterization: The low-energy Lagrangian is defined with generic anomaly vectors for QCD ($N$) and QED ($E$). The axion potential is decomposed into a QCD-induced term and a generic PQ-breaking potential ($V_{PQ}$).
2. Classification of Breaking Sources: The PQ-breaking sources are classified into two sets: those fully aligned with the QCD anomaly vector ($\mathcal{C}$) and those non-fully-aligned ($\mathcal{A}$). This allows the mass matrix to be parameterized as $M^2 = \bar{\Lambda}_{QCD}^4 N N^T + M_A^2$, where $\bar{\Lambda}_{QCD}$ captures the effective QCD scale (modified by aligned breaking) and $M_A^2$ encodes non-aligned breaking effects.
3. Relaxation of Assumptions: The analysis explicitly relaxes two standard assumptions:
   • The existence of a PQ symmetry broken exclusively by QCD (allowing for "general-PQ" systems where all symmetries are broken beyond QCD, yet a CP-conserving minimum exists).
   • The universality of the $E/N$ ratio (allowing for non-universal anomaly coefficients).
4. Derivation of Sum Rules: Using the Sherman-Morrison formula for matrix inversion, the authors derive a general sum rule for the photon couplings and masses of the axion eigenstates. This rule connects experimental observables to the underlying structure of PQ breaking and anomaly alignment.
5. Case Studies: The framework is applied to two-axion systems to identify distinct phenomenological regimes and then generalized to $N$-axion systems. Numerical scans are performed to visualize the parameter space.

Key Contributions and Results

• Generalized QCD Line: The authors identify that in general-PQ systems, the effective QCD scale $\bar{\Lambda}_{QCD}$ can differ from the standard $\Lambda_{QCD}$ due to interference with other PQ-breaking sources. This shifts the "QCD line" itself, potentially moving the canonical axion solution to the left or right of the traditional band.
• Phenomenological Regimes: The paper categorizes multi-axion scenarios based on the hierarchy of breaking scales ($\Lambda_i$) relative to $\bar{\Lambda}_{QCD}$ and the alignment of anomaly vectors:
  • Right of the Band: Axions can lie to the right of the QCD band if non-QCD PQ-breaking scales ($\Lambda_i$) are significantly larger than $\Lambda_{QCD}$. This occurs naturally in models where QCD emerges as a diagonal subgroup of a larger strong sector.
  • Left of the Band: Axions can appear to the left of the QCD band (lighter and/or more strongly coupled than the canonical relation) if:
    1. The anomaly coefficients are non-universal ($E_i/N_i \neq E_j/N_j$), causing the photon coupling vector to misalign with the gluon anomaly vector.
    2. The effective QCD scale is suppressed ($\bar{\Lambda}_{QCD} \ll \Lambda_{QCD}$), which can occur in mirror world constructions with destructive interference.
  • Photophobic States: If anomaly ratios are universal, light axions orthogonal to the QCD direction become "photophobic" (suppressed photon coupling), making them difficult to detect.
• General Sum Rule: A new sum rule is derived:
  $$\sum_i \frac{\tilde{g}_{a_i\gamma}^2}{m_i^2} = \left(\frac{\alpha_{em}}{2\pi}\right)^2 \frac{1}{1+r} \left[ \sum_i \left(\frac{E_i}{N_i} - C_\chi\right)^2 \frac{\Lambda_{QCD}^4}{\Lambda_i^4} + \sum_{i<j} \frac{\Lambda_{QCD}^4 \bar{\Lambda}_{QCD}^4}{\Lambda_i^4 \Lambda_j^4} \left(\frac{E_i}{N_i} - \frac{E_j}{N_j}\right)^2 \right]$$
  where $r$ depends on the breaking scales. This rule generalizes previous results (e.g., Ref. [55]) and shows that:
  • A sum rule value $< 1$ implies either undiscovered axions or a lack of a pure QCD-PQ direction.
  • A sum rule value $> 1$ implies non-universal anomaly coefficients or specific UV dynamics (e.g., mirror worlds).
• UV Completions: The paper maps these phenomenological regimes to specific UV-complete theories, including:
  • Mirror Worlds: Can generate axions to the left of the band with universal coefficients.
  • Diagonal Subgroup Embeddings: Can generate axions to the right of the band.
  • String Theory/Extra Dimensions: Naturally predict non-universal $E/N$ ratios, allowing for axions to the left of the band even with universal coefficients in specific limits.

Significance and Claims
The paper claims that the multi-axion paradigm offers a substantially broader and phenomenologically richer landscape for solving the Strong CP problem than previously appreciated. By moving beyond the restrictive assumptions of QCD-PQ systems and universal anomaly coefficients, the authors demonstrate that:

1. Experimental Reach: Axion searches must extend beyond the canonical QCD band. Axions to the left of the band are within the reach of current and future experiments, while those to the right may require new search strategies.
2. UV Diagnostics: The discovery of a multi-axion system, characterized by the general sum rule and the specific location of axions in parameter space, can provide direct insight into the ultraviolet completion of the Standard Model. For instance, observing axions to the left of the band with non-universal $E/N$ ratios would rule out simple Grand Unified Theories (GUTs) like $SU(5)$ or $SO(10)$, which predict universal ratios, and point towards more complex embeddings (e.g., Pati-Salam, Trinification, or string theory constructions).
3. Theoretical Completeness: The work provides a systematic framework to interpret potential multi-axion signals, distinguishing between generic Axion-Like Particles (ALPs) and axions arising specifically from solutions to the Strong CP problem.

The authors conclude that a combined experimental effort targeting these extended parameter spaces is motivated by strong theoretical grounds and is essential for a complete understanding of fundamental physics and the UV structure of the SM.