QCD Axion Model and SU(3)f Embedding

• The QCD axion model is a theoretical framework that introduces a pseudo-Nambu–Goldstone boson via the Peccei–Quinn mechanism to dynamically resolve the strong CP problem.
• It embeds the dangerous PQ discrete subgroup into a continuous SU(3)f symmetry, effectively resolving the cosmological domain wall problem.
• The model predicts measurable phenomena such as dark radiation contributions, rare kaon decays, and stringent constraints on the axion decay constant.

The QCD axion model refers to a broad class of theoretical proposals in which the axion—a pseudo-Nambu–Goldstone boson associated with the spontaneous breaking of a global Peccei–Quinn (PQ) symmetry—solves the strong CP problem of quantum chromodynamics (QCD) by dynamically relaxing the QCD θ-parameter. Over decades of development, numerous models have been constructed to address not only the original CP problem but also a series of cosmological, astrophysical, and theoretical challenges, including the domain wall problem, the axion "quality problem," and the axion’s viability as dark matter. Below, the QCD axion model is presented in the context of modern theoretical developments and phenomenological consequences, as delineated in the referenced literature.

1. Peccei–Quinn Mechanism and Axion Emergence

The Peccei–Quinn mechanism postulates a spontaneously broken anomalous global U(1)$_{\rm PQ}$ symmetry, under which quark fields transform nontrivially. Upon breaking, the axion $a$ appears as a pseudo-Nambu–Goldstone boson. The essential coupling responsible for solving the strong CP problem is

$$\mathcal{L}_{aGG} = \frac{\alpha_s}{8\pi} \frac{a}{F_a} G_{\mu\nu}^a \tilde{G}^{a\mu\nu},$$

where $F_a$ is the axion decay constant. The axion potential is generated nonperturbatively by QCD instantons, naturally minimizing at the CP-conserving value, thereby driving the effective θ-angle $\bar{\theta} = 0$. In generic models, $F_a$ is undetermined, but cosmological and astrophysical constraints, as well as nontrivial model-building requirements, restrict its value to specific ranges (typically $10^9$–$10^{12}$ GeV for "invisible" axions, but alternative constructions exist).

A further consequence is the axion mass, determined by the QCD topological susceptibility,

$$m_a^2 = \frac{m_u m_d}{(m_u + m_d)^2} \frac{m_\pi^2 f_\pi^2}{F_a^2}[1 + \text{NLO chiral corrections}],$$

with higher-order effects and nonanalytic corrections incorporated at subpercent levels in precision computations (Cortona et al., 2015).

2. Domain Wall Problem and Discrete Symmetry Embedding

The original axion models encountered the cosmological domain wall problem: at the QCD phase transition, $U(1)_{\rm PQ}$ breaking leads to a discrete subgroup, resulting in $N_{\rm DW} > 1$ degenerate vacua and cosmologically catastrophic domain walls. The embedding of the dangerous discrete subgroup into a continuous non-Abelian global symmetry, as realized in the variant with a global $SU(3)_f$ flavor symmetry (Kawasaki et al., 2015), invokes the Lazarides–Shafi mechanism: the discrete residual $Z_3$ of $U(1)_{\rm PQ}$ is identified with the center of $SU(3)_f$. The PQ-breaking field, $x_0 = \mathrm{diag}(v_1, v_2, v_3)$, breaks both PQ and family symmetry simultaneously. As a result, all vacua are continuously connected, and the effective domain wall number is reduced to $N_{\rm DW} = 1$, ensuring domain wall networks decay and avoiding cosmological disasters.

The embedding can be summarized in the transformation properties: $$Q \rightarrow e^{2\pi i n/3} Q, \quad x \rightarrow e^{2\pi i n/3} x,$$ with $n \in \mathbb{Z}$ and the center of $SU(3)_f$ matching the discrete PQ remnant. This mechanism is directly generalizable to other choices of non-Abelian continuous family symmetry absorbing the PQ discrete remnant, provided the centers match the required subgroup structure.

3. Phenomenological Signatures: Dark Radiation and Exotic Decays

The extension of $U(1)_{\rm PQ} \times SU(3)_f$ produces not just the axion but also a family of eight additional Nambu–Goldstone (NG) bosons—familons. When PQ and $SU(3)_f$ break, both the axion and familons are produced. While cold axions can constitute dark matter via the misalignment mechanism, thermally produced axion and familon populations decouple at high temperatures ($T \sim 10^9$–$10^{10}$ GeV), remaining relativistic and contributing to dark radiation. The effective number of additional neutrino species is thus shifted: $$\Delta N_{\text{eff}}^\text{axion} \simeq 0.027, \qquad \Delta N_{\text{eff}}^\text{total} \simeq 0.24,$$ where the familon contribution dominates. This is within current CMB constraints but will be subject to future scrutiny (e.g., by CMB polarization measurements with sensitivity to $\Delta N_\text{eff} \sim 0.016$) (Kawasaki et al., 2015).

Additional consequences arise from flavor-changing interactions of the familons. The familon–quark derivative coupling

$$\frac{1}{F_f} (\partial_\mu \phi) \left[\bar{d} \gamma^\mu s\right] + \mathrm{h.c.}$$

induces exotic kaon decays, notably $K^+ \to \pi^+ + f$. The decay rate is suppressed by $F_f^{-2}$, and current null observations set a robust lower bound,

$F_f \gtrsim 5.3 \times 10^{11} \text{ GeV}$

for the familon decay constant, compatible (marginally) with constraints on $F_a$ from dark matter abundance. Upcoming kaon decay experiments (e.g., NA62) will further probe this regime.

4. Constraints on the Axion Decay Constant and Cosmological Viability

The axion decay constant in this framework is connected to the symmetry breaking VEVs,

$$\frac{1}{F_a} \simeq \frac{1}{v_1} + \frac{1}{v_2} + \frac{1}{v_3},$$

and viable dark matter abundance is found when $F_a \sim (0.3$–$1.5)\times 10^{11}$ GeV, balancing misalignment and topological defect decay contributions. The requirement that $F_f \gtrsim 5.3 \times 10^{11}$ GeV (from kaon decay) is close to tension but can be accommodated for appropriate VEV assignments. This places strong constraints on the symmetry-breaking scales and impacts the parameter space for the model.

Furthermore, the presence of additional relativistic degrees of freedom just below existing bounds imposes a target for future cosmological experiments that could directly test this class of axion models.

5. Model-Theoretical and Cosmological Implications

The $SU(3)_f$-embedded QCD axion model has several far-reaching implications:

• Resolution of the domain wall problem: Embedding the PQ discrete subgroup in $SU(3)_f$ removes the domain wall multiplicity, enabling cosmological safety even with post-inflation PQ symmetry breaking, assuming a three-family structure.
• Portal to precision cosmology: The predicted extra dark radiation provides a direct, testable prediction for future CMB measurements of $N_\text{eff}$. Any deviation consistent with $\Delta N_\text{eff} \simeq 0.24$ would strongly support such symmetry structures.
• Flavor physics connection: Flavor-changing familon couplings offer rare kaon decay signals closely connected to the underlying symmetry breaking. Nonstandard branching fractions in $K^+ \to \pi^+ +$ (invisible) final states would point to familon contributions.
• Integration with broader beyond-the-standard-model (BSM) questions: The framework naturally incorporates the seesaw mechanism for neutrino masses, with right-handed neutrinos charged under $SU(3)_f$, and can enable baryogenesis via thermal leptogenesis, coupling together several key BSM puzzles.

A table summarizing distinctive features and observables is presented below.

Feature	Prediction	Experimental Test
Dark radiation	$\Delta N_{\rm eff}\simeq 0.24$	Future CMB experiments
Exotic kaon decay	$K^+\to \pi^+ + f$, $F_f \gtrsim 5.3\times 10^{11}$ GeV	NA62, CERN
Axion decay constant	$F_a \sim (0.3$–$1.5)\times 10^{11}$ GeV	Axion haloscope, astrophysical constraints
Resolution of domain walls	$N_{\rm DW}=1$ by $SU(3)_f$ embedding	Cosmological consistency

6. Broader Context and Outlook

Embedding the PQ discrete symmetry into a continuous family symmetry represents a significant evolution of axion model building, addressing the long-standing domain wall problem and relating axion properties to family structure. The model creates a tight interplay between early universe cosmology, axion dark matter, flavor physics, and possible neutrino sector extensions.

Current and forthcoming experiments, particularly those sensitive to rare kaon decays and precision CMB measurements, have the potential to decisively test this framework. Should $\Delta N_{\rm eff}$ deviate upward by the predicted amount, or rare kaon decays display nonstandard rates, this would provide strong evidence for the presence of both axion and familon sectors of the type described here.

In summary, the $SU(3)_f$-embedded QCD axion model represents a cosmologically safe and predictive realization of the axion mechanism, deriving phenomenological consequences for both cosmology and flavor physics directly from the required structure that resolves the domain wall problem (Kawasaki et al., 2015).