QCD axion from chiral gauge theories

This paper presents calculable supersymmetric chiral gauge theory models where the Peccei-Quinn symmetry is spontaneously broken by non-perturbative dynamics, leading to a QCD axion scenario compatible with SU(5) grand unification that requires the GUT and PQ breaking scales to coincide with a SUSY breaking scale of approximately $10^9$ GeV.

The Gist

The Big Mystery: The "Ghost" in the Machine

Imagine the universe is a giant, complex clockwork machine. For decades, physicists have noticed a strange glitch in one of its gears: the Strong CP Problem.

In the Standard Model (our current rulebook for physics), there is a setting called the $\theta$-angle. Think of this like a volume knob for a specific type of cosmic noise. If this knob were turned up even a tiny bit, it would cause neutrons (the building blocks of atoms) to act like tiny magnets, spinning in a way that creates a detectable "electric dipole moment."

But here's the mystery: The knob is set to zero. Neutrons don't show this magnetic spin. The universe is perfectly silent in this regard. Why is this knob so perfectly tuned to zero? It's like finding a radio that is perfectly tuned to a station with no static, without anyone ever touching the dial.

The Hero: The Axion

To solve this, physicists proposed a hero particle called the Axion.

• The Analogy: Imagine the $\theta$-angle isn't a fixed knob, but a ball sitting on a hilly landscape. The ball naturally rolls down to the lowest point (the valley), which happens to be exactly zero.
• The Axion is the name of that ball. It's a particle that moves around, constantly adjusting the "knob" until it finds the perfect zero spot, solving the mystery of why the universe is so quiet.

The Problem: Where does the Axion come from?

Usually, we think the Axion gets its power from a "symmetry breaking" event, like a ball rolling down a hill because the hill was built that way. But in this paper, the authors ask: What if the hill is built by the universe itself?

They propose that the Axion isn't just a fundamental particle we add to the list; it's a composite particle.

• The Analogy: Imagine you have a bag of marbles (fundamental particles). If you shake the bag hard enough (strong dynamics), the marbles might stick together to form a new, larger shape. The Axion is that new shape. It's a "molecule" made of other particles, held together by a force so strong we can't see the individual marbles anymore.

The New Toolkit: Supersymmetry (SUSY)

Studying these "sticky marbles" is incredibly hard. Usually, the math breaks down because the forces are too strong.

• The Analogy: It's like trying to predict the weather during a hurricane using a paper airplane model. The model just falls apart.

The authors use a special trick called Supersymmetry (SUSY).

• The Analogy: SUSY is like giving the hurricane a pair of glasses that makes the wind behave in a predictable, orderly way. It allows the authors to do the math on these "sticky marbles" without the equations exploding. They found that in these specific "chiral" theories (a fancy type of particle interaction), the universe naturally builds the Axion hill, and the ball rolls down to zero automatically.

The Grand Unification: The SU(5) Puzzle

The authors didn't just stop at building an Axion; they tried to fit it into a bigger picture called Grand Unification (GUT).

• The Analogy: Imagine the Standard Model has three different languages: one for the strong force, one for the weak force, and one for electromagnetism. Physicists believe that at very high energies (like right after the Big Bang), these three languages were actually just one single "Universal Language."
• The authors built a model where the Axion and this "Universal Language" (specifically the SU(5) model) get along perfectly.

The "Goldilocks" Result

When they ran the numbers to see if their model works, they found a very specific set of conditions, like a "Goldilocks" scenario:

1. The Scale: The moment the Axion is born (PQ breaking) and the moment the "Universal Language" splits back into three (GUT scale) happen at the exact same time. They are twins.
2. The Mass: The "supersymmetric" particles (the helpers that make the math work) need to have a very specific weight. They can't be too light, or the math fails. They can't be too heavy, or the universe breaks. The authors found they need to be around $10^9$ GeV (a billion times heavier than a proton).
3. The Proton: A major worry in these theories is that protons might decay (fall apart). The authors checked this and found that while protons could decay, it would take so long (trillions of trillions of years) that our current experiments won't see it yet. However, future, super-sensitive detectors (like Hyper-Kamiokande) might just be able to catch a glimpse of it.

Why Should We Care?

This paper is exciting because it offers a self-contained solution.

• Instead of just saying "Let's add an Axion to fix the problem," they showed a mechanism where the Axion is a natural byproduct of the universe's deepest forces.
• It connects the mystery of the neutron (Strong CP) with the mystery of the universe's structure (Grand Unification).
• It gives us a specific target for future experiments. If we build a detector sensitive enough to see protons decaying at a specific rate, we could prove this theory right.

Summary in One Sentence

The authors used a special mathematical trick (Supersymmetry) to show that the universe naturally builds a "fix-it" particle (the Axion) out of strong forces, solving a decades-old mystery about why neutrons don't act like magnets, all while fitting perfectly into a theory where all forces were once one.

Technical Summary

1. Problem Statement

The paper addresses two major open problems in particle physics:

• The Strong CP Problem: The Standard Model (SM) allows for a CP-violating term in the QCD Lagrangian ($\theta G\tilde{G}$), yet experimental limits on the neutron electric dipole moment constrain the effective angle $|\bar{\theta}| \lesssim 10^{-10}$. The QCD axion is the leading solution, requiring a global $U(1)_{PQ}$ symmetry that is spontaneously broken and anomalous under QCD.
• The Axion Quality Problem: In many axion models, the $U(1)_{PQ}$ symmetry is an accidental symmetry that can be broken by higher-dimensional operators (e.g., from quantum gravity), reintroducing the strong CP problem.
• Theoretical Challenge: While strong dynamics in QCD-like theories are well-studied, constructing axion models based on chiral gauge theories is difficult due to the lack of analytical tools for non-perturbative dynamics.

The authors aim to construct a QCD axion model where the PQ symmetry is spontaneously broken by the non-perturbative dynamics of a chiral gauge theory, utilizing Supersymmetry (SUSY) to make the dynamics calculable, and embedding this into a Grand Unified Theory (GUT) framework.

2. Methodology

The authors employ the following theoretical framework and techniques:

• Supersymmetric Chiral Gauge Theories: They utilize SUSY to analyze the non-perturbative dynamics of chiral gauge theories. In the limit of small soft SUSY breaking, the dynamics can be solved using holomorphic superpotentials and chiral Lagrangians.
• Model Construction:
  • Toy Models: They first construct two explicit "toy" models based on chiral gauge groups:
    1. SUSY Georgi-Glashow type: Based on $SU(2N+4)_1 \times SU(N)_2$.
    2. SUSY Bars-Yankielowicz type: Based on $SU(2N-4)_1 \times SU(N)_2$.
  • Mechanism: In these models, the PQ symmetry is broken by the vacuum expectation values (VEVs) of composite operators generated by the strong dynamics of the chiral gauge sector. The axion arises as a pseudo-Nambu-Goldstone boson (pNGB).
• GUT Embedding: They construct a specific phenomenological model based on the Georgi-Glashow type with $N=5$, resulting in an $SU(14)_1 \times SU(5)_2$ gauge symmetry. This is designed to be compatible with $SU(5)$ Grand Unification.
• Renormalization Group (RG) Analysis: They perform a two-loop RG analysis of gauge couplings and one-loop analysis of the top Yukawa coupling to test gauge coupling unification.
• Phenomenological Constraints: The model is tested against:
  • The observed Higgs boson mass ($125$ GeV).
  • Proton decay limits (specifically $p \to \pi^0 e^+$).
  • Cosmological constraints (domain walls, dark matter overclosure).

3. Key Contributions and Results

A. Theoretical Construction of Axion Models

• Dynamical PQ Breaking: The authors demonstrate that in SUSY chiral gauge theories, the vacuum is stabilized by soft SUSY breaking masses, leading to non-zero VEVs for chiral multiplets. This spontaneous breaking generates the axion.
• Domain Wall Number ($N_{DW}$): They calculate the domain wall number for their models. For the Georgi-Glashow type, $N_{DW} = 4(N+1)$. For the specific GUT model ($N=5$), $N_{DW} = 24$. This implies a potential domain wall problem, which they address via cosmological assumptions (low reheating temperature).
• Axion Decay Constant ($f_a$): The decay constant is derived as a function of the dynamical scale ($\Lambda$) and the soft SUSY breaking mass ($m$). In the GUT model, $f_a$ is naturally tied to the GUT scale.

B. The GUT-Motivated QCD Axion Model

The authors propose a specific model with the following features:

• Gauge Group: $SU(14)_1 \times SU(5)_2 \times U(1)_{PQ} \times \dots$
• Matter Content: Includes MSSM fields embedded in $SU(5)_2$ and chiral multiplets ($\bar{F}_q, \bar{F}_{\bar{q}}, A$) responsible for PQ breaking.
• Symmetry Breaking Pattern:
  • $SU(14)_1 \times SU(5)_2$ breaks to $SU(5)_{GUT}$ at the PQ scale ($M_{PQ}$).
  • $SU(5)_{GUT}$ breaks to the SM gauge group at the GUT scale ($M_{GUT}$).

C. Numerical Analysis and Unification

• Coupling Unification: The authors find that successful gauge coupling unification requires a specific hierarchy: $M_{PQ} \approx M_{GUT}$.
  • If $M_{PQ} < M_{GUT}$, the running couplings hit a Landau pole below the unification scale, preventing unification.
  • If $M_{PQ} = M_{GUT}$, the couplings unify successfully.
• SUSY Breaking Scale: To satisfy the 125 GeV Higgs mass constraint and avoid Landau poles, the model requires a mini-split SUSY spectrum:
  • Sfermion masses: $M_S \sim \mathcal{O}(10^9)$ GeV.
  • pNGB masses: $M_{pNGB} \sim \mathcal{O}(10^{10})$ GeV.
  • Gaugino masses: $M_{1/2} \sim \mathcal{O}(10^6)$ GeV (generated via Anomaly Mediated SUSY Breaking, AMSB).
• Proton Decay:
  • Dimension-5 operators are suppressed by heavy sfermions ($10^9$ GeV).
  • Dimension-6 operators (mediated by $X, Y$ gauge bosons) are the dominant constraint.
  • For the benchmark case $(a,b)=(2,3)$ in the unification analysis, the predicted proton lifetime is $\tau_p \approx 5.4 \times 10^{36}$ years, which is safely above current experimental limits ($2.4 \times 10^{34}$ years) and within the reach of future experiments like Hyper-Kamiokande.

D. Cosmology and Stability

• Domain Wall Problem: With $N_{DW}=24$, the model risks overclosing the universe with stable domain walls. The authors propose that the reheating temperature must be lower than the pNGB mass ($T_R < 10^{10}$ GeV) to avoid this.
• Axion Dark Matter: The axion decay constant is near the GUT scale ($f_a \sim 10^{16}$ GeV), which is higher than the canonical value for the misalignment mechanism ($10^{12}$ GeV). This requires either a suppressed initial misalignment angle or entropy production to dilute the axion density.
• Small Instantons: The contribution of small instantons from the $SU(5)_2$ sector to the axion mass is calculated to be negligibly small ($\sim 10^{-105}$) due to strong suppression factors involving gaugino masses and fermion contractions.

4. Significance

• New Class of Axion Models: This work provides the first explicit, calculable examples of QCD axions arising from chiral gauge theories, expanding the landscape beyond QCD-like confining theories.
• SUSY as a Tool: It demonstrates the utility of SUSY in solving non-perturbative dynamics of chiral theories, allowing for precise predictions of the axion decay constant and vacuum structure.
• GUT Compatibility: The model successfully embeds the axion mechanism into an $SU(5)$ GUT framework without spoiling gauge coupling unification, provided the PQ breaking scale coincides with the GUT scale.
• Testability: The model makes specific predictions for the proton lifetime and the SUSY spectrum (mini-split), offering clear targets for future proton decay experiments and indirect cosmological observations.

In conclusion, the paper establishes a robust theoretical framework where the strong CP problem is solved by the dynamics of a chiral gauge theory, naturally linking the axion scale to the GUT scale while maintaining consistency with current experimental bounds on proton decay and the Higgs mass.