Emergent axion and Higgs boson from strong dynamics

This paper proposes a unified composite Higgs model based on an $SU(4)/Sp(4)$ coset that simultaneously solves the hierarchy and strong CP problems by identifying the axion with a CP-odd singlet whose mass is enhanced by a hidden gauge sector, allowing for a viable GeV-scale axion accessible at collider experiments.

The Gist

Imagine the universe is a giant, complex machine built by a master engineer. For decades, we've been trying to understand how this machine works using a manual called the Standard Model. This manual is incredibly accurate; it explains how light, magnets, and atomic nuclei behave. But, like any old manual, it has two glaring problems that don't make sense:

1. The "Too Heavy" Problem (Hierarchy Problem): The manual says the "Higgs boson" (the particle that gives everything mass) should be incredibly heavy, like a bowling ball. But in reality, it's light, like a feather. Why is it so light? The manual implies it should be heavy, and keeping it light requires "fine-tuning" the universe to a ridiculous degree of precision.
2. The "Ghostly Mirror" Problem (Strong CP Problem): The manual says that if you look at the strong nuclear force (which holds atoms together) in a mirror, it should look exactly the same. But the math suggests it shouldn't. It's like a mirror that flips left and right but refuses to flip up and down. We don't see this in nature, but the math says it should happen.

For years, physicists have tried to fix these two problems separately. This paper proposes a clever, unified solution: What if the Higgs boson and the "Axion" (the particle that fixes the mirror problem) are actually siblings born from the same family?

Here is the story of their solution, explained with some everyday analogies.

1. The Composite House (The Higgs)

Imagine the Higgs boson isn't a fundamental, indivisible particle. Instead, think of it as a house built from bricks.

• In this model, the "bricks" are new, tiny particles held together by a super-strong glue called Hypercolor.
• Because the house is made of these bricks, it naturally stays light and stable without needing that weird "fine-tuning." This solves the "Too Heavy" problem.
• The paper uses a specific blueprint (mathematically called an SU(4)/Sp(4) coset) to build this house. Interestingly, this blueprint naturally produces not just the house (Higgs), but also a strange, invisible "ghost" particle hiding in the attic. This ghost is the Axion.

2. The Ghost in the Attic (The Axion)

Usually, an Axion is a particle that solves the "Ghostly Mirror" problem by gently nudging the universe until the mirror looks right again.

• The Catch: Standard Axions are usually very weak and very light. If they were this light, they would have been detected by now. But we haven't seen them.
• The Paper's Twist: The authors say, "What if we make this Axion heavier and give it a little more muscle?"
• They do this by introducing a new, hidden garage (a hidden gauge sector) next to the house. This garage has its own super-strong glue (confinement scale) that is even stronger than the one holding the house together.
• This extra strength makes the Axion heavier (around the mass of a heavy atom, in the "GeV" range), which allows it to hide from current experiments while still doing its job of fixing the mirror problem.

3. The Grand Unification (The "Grandcolor")

Here is the most elegant part of the story.

• We have the "House Glue" (Hypercolor) and the "Garage Glue" (Hidden Color).
• The paper proposes that at a very high energy level, these two glues were actually one single, giant glue called Grandcolor.
• Think of it like a river splitting into two streams. Because they came from the same source, they share the same "water level" (topological angle).
• This ensures that the Axion fixes the mirror problem for both the visible universe and the hidden garage simultaneously. It's a perfect handshake between the two sectors.

4. Why This Matters (The Phenomenology)

The paper isn't just math; it predicts what we might find in a particle collider (like the Large Hadron Collider).

• The "Goldilocks" Zone: They found a specific range of weights for this Axion (around 1 to 10 GeV) where it is heavy enough to avoid old experiments but light enough to be created in new ones.
• The Hunt: If we look at rare decays of heavy particles (like B-mesons) in future experiments, we might see this Axion popping out. It would be like finding a specific, rare coin in a pile of change.
• The Connection: If we find this particle, it proves that the Higgs and the Axion are indeed related, solving two of physics' biggest mysteries with one single mechanism.

The Big Picture Analogy

Imagine you are trying to fix a broken clock (the Universe).

• Problem A: The clock's spring is too loose (Higgs is too light).
• Problem B: The clock's hands are spinning backward in the mirror (Strong CP violation).
• Old Solutions: You tried to fix the spring with a wrench and the hands with a screwdriver, but the tools didn't fit.
• This Paper's Solution: You realize the spring and the hands are actually part of the same gear mechanism. By adjusting the main gear (the Grandcolor symmetry), you tighten the spring and straighten the hands at the same time. Plus, you add a small weight to the gear so it spins at the right speed to avoid being noticed by the clock inspector (experimental bounds).

Conclusion

This paper suggests that the universe is more interconnected than we thought. The particle that gives us mass (Higgs) and the particle that keeps the laws of physics symmetrical (Axion) are likely two sides of the same coin, born from a deeper, more fundamental layer of reality. If we can find this "heavy" Axion in the coming years, it would be a massive victory for our understanding of how the universe is built.

Technical Summary

Here is a detailed technical summary of the paper "Emergent axion and Higgs boson from strong dynamics" by F. Goertz and A. Incrocci.

1. Problem Statement

The paper addresses two fundamental puzzles in the Standard Model (SM):

• The Hierarchy Problem: The instability of the Higgs mass against quantum corrections, requiring extreme fine-tuning if the SM is valid up to high scales.
• The Strong CP Problem: The absence of CP violation in Quantum Chromodynamics (QCD), quantified by the $\theta$-angle, which is experimentally constrained to be nearly zero ($\bar{\theta} < 10^{-10}$) despite no symmetry in the SM forbidding it.

The Conflict:
A unified solution often proposes that the Higgs boson and the axion (the solution to the Strong CP problem) arise from the same symmetry breaking mechanism. In such models, the axion decay constant ($f_a$) is identified with the Higgs compositeness scale ($f$).

• Constraint 1 (Hierarchy): To solve the hierarchy problem naturally, $f$ must be close to the electroweak scale ($v \approx 246$ GeV), typically $f \sim \text{TeV}$.
• Constraint 2 (Axion): Standard QCD axion phenomenology requires $f_a \gtrsim 10^8$ GeV to satisfy experimental bounds (e.g., from $K^+ \to \pi^+ a$ decays).
• The Challenge: Reconciling a low $f_a$ (for the hierarchy) with the need to evade experimental bounds on axion couplings, while simultaneously solving the Strong CP problem.

2. Methodology and Model Setup

The authors propose a unified framework based on strong dynamics involving a minimal fundamental composite Higgs model extended with a hidden gauge sector.

A. Gauge Structure and Symmetry Breaking

• Grandcolor (GC) Sector: The model introduces an enlarged non-Abelian gauge group, $SU(2N+3)_{GC}$, which unifies the SM color group $SU(3)_c$ and a new hidden confining group $Sp(2N)$ (or $Sp(2N_{GC})$).
  • Breaking Pattern: $SU(2N+3)_{GC} \xrightarrow{\langle \Phi \rangle, \langle \Xi \rangle} SU(3)_c \times Sp(2N)$.
  • This unification ensures that the topological angles ($\theta_c$ and $\theta_{Sp}$) of the two sectors are identical at the unification scale $f_{GC}$, satisfying the condition $\bar{\theta}_c = \bar{\theta}_{Sp}$ at tree level.
• Hypercolor (HC) Sector: A separate confining gauge group, $G_{HC}$ (specifically $SO(N_{HC})$ or $Sp(2N_{HC})$), is introduced to generate the composite Higgs.
  • Coset: The model utilizes the minimal $SU(4)/Sp(4)$ coset.
  • Field Content: Fermions $\omega$ (charged under HC and EW) and $\chi$ (charged under HC and GC) form condensates.
  • Pseudo-Nambu-Goldstone Bosons (pNGBs): The breaking of global symmetries in the HC sector produces a scalar bidoublet (identified as the Higgs) and a CP-odd scalar singlet ($\eta$), which is identified as the axion candidate.

B. Mechanism for Mass Enhancement

To resolve the tension between a low $f_a$ and experimental bounds, the authors employ a mechanism where the axion mass receives contributions from two confining sectors:

1. QCD Sector: Standard contribution $\Lambda_{QCD}^4$.
2. Hidden Sector: A new contribution $\Lambda_{Sp}^4$ from the $Sp(2N)$ group, where the confinement scale $\Lambda_{Sp}$ is significantly larger than $\Lambda_{QCD}$ (and potentially larger than the EW scale).

The axion potential is approximately:
$$m_a^2 f_a^2 \simeq \Lambda_{QCD}^4 + \Lambda_{Sp}^4$$
Since $\Lambda_{Sp} > \Lambda_{QCD}$, the axion mass is dominated by the hidden sector, allowing $f_a$ to remain low (TeV scale) while $m_a$ becomes heavy enough to evade flavor constraints (specifically $K \to \pi a$).

C. Control of CP Violation

A critical requirement is ensuring that radiative corrections do not misalign the vacuum such that $\bar{\theta}_c \neq \bar{\theta}_{Sp}$.

• The authors demonstrate that tree-level unification via Grandcolor guarantees $\bar{\theta}_c = \bar{\theta}_{Sp}$.
• They analyze radiative corrections from Yukawa couplings and scalar potentials, showing that with appropriate flavor symmetries and parameter choices, these misalignments can be suppressed below the $10^{-10}$ threshold required to solve the Strong CP problem.

3. Key Contributions

1. Unified Origin: The paper demonstrates that the CP-odd singlet in the minimal $SU(4)/Sp(4)$ composite Higgs model can naturally serve as the axion, solving both the Hierarchy and Strong CP problems simultaneously.
2. Heavy Axion via Hidden Sector: It introduces a specific realization where the axion mass is boosted by a hidden $Sp(2N)$ sector with a confinement scale $\Lambda_{Sp} > \Lambda_{QCD}$. This allows the compositeness scale $f$ to be as low as a few TeV without violating axion experimental bounds.
3. Stability of the Solution: The authors provide a rigorous analysis of the axion potential, proving that the alignment of the topological angles is stable against radiative corrections and that the vacuum does not spontaneously break CP.
4. Phenomenological Viability: They identify a viable parameter space where the axion mass is in the GeV range ($m_a \sim 10$ GeV), making it accessible to current and future collider experiments and flavor physics observables.

4. Results

• Axion Mass and Decay Constant:
  • The model allows for a decay constant $f_a \in [2, 10^3]$ TeV.
  • The axion mass is generated primarily by the hidden sector: $m_a \sim \sqrt{y_{\psi} y_{\psi}'} \frac{f_{Sp}^2}{f_a}$.
  • For benchmark parameters ($f_{Sp} \sim \text{TeV}$, $f_a \sim 10^3$ TeV), the axion mass can reach $m_a \sim 10$ GeV.
• Couplings:
  • The axion ($\eta$) couples to SM fermions (specifically the top quark) via Partial Compositeness.
  • Tree-level couplings to gluons are absent (since the $\omega$ fermions are color-neutral), but effective couplings are generated at the loop level.
  • The coupling to photons is suppressed, avoiding strong constraints from beam-dump experiments for $m_a > 0.5$ GeV.
• Constraints:
  • Flavor Physics: The most stringent constraints come from rare meson decays ($B \to K a$ and $K \to \pi a$). The model is viable if the Yukawa couplings of the exotic fermions are suppressed or if the axion mass is sufficiently high ($> 1$ GeV).
  • Higgs Mixing: The model requires the mixing between the Higgs and the "Higgs-like" pNGBs from the hidden sector to be small ($\theta_{mix} \ll 1$). This is achieved if the top-partner mass in the hidden sector is sufficiently heavy.
• Parameter Space:
  • A "natural" region exists where $f_a \sim 10^3$ TeV and $m_a \sim 10$ GeV. While $f_a$ is higher than the strict TeV scale, it is significantly lower than the $10^8$ GeV required for standard axions, offering a partial solution to the hierarchy problem with moderate tuning.
  • If $f_a \sim 2$ TeV, the axion mass must be heavier or couplings must be tuned to avoid flavor bounds, potentially reintroducing fine-tuning.

5. Significance

This work provides a compelling UV-complete framework where the solutions to the two most prominent naturalness puzzles are intrinsically linked.

• Theoretical Impact: It validates the idea that the Strong CP problem can be solved by a "heavy" axion emerging from the same dynamics that generate the Higgs, challenging the paradigm that axions must be extremely light and weakly coupled.
• Experimental Impact: Unlike standard axion models which are often invisible to colliders, this model predicts a GeV-scale axion with couplings to the top quark. This makes it a prime target for:
  • LHC and Future Colliders: Searches for $pp \to a \to \gamma\gamma$ or $a \to \text{invisible}$, and rare Higgs decays.
  • Flavor Experiments: Precision measurements of $B$ and $K$ meson decays (e.g., at Belle II, LHCb).
• Naturalness: It offers a path to a composite Higgs with a compositeness scale in the multi-TeV range, significantly reducing the fine-tuning required compared to the SM, while dynamically solving the Strong CP problem without invoking mirror worlds or discrete symmetries that are difficult to embed in a UV-complete theory.

In summary, the paper successfully constructs a "Grand Color" composite Higgs model where the emergent axion is heavy enough to evade current bounds yet light enough to be discovered at colliders, providing a unified and testable solution to the hierarchy and Strong CP problems.