PQWW Axion Model Overview

• The PQWW axion model is a two-Higgs-doublet extension of the Standard Model that introduces a pseudo-Nambu-Goldstone boson to cancel the strong CP violating parameter.
• The model utilizes a global U(1)PQ symmetry, whose spontaneous breaking at the electroweak scale fixes axion couplings to gluons, photons, and fermions via chiral anomalies.
• Experimental and astrophysical constraints decisively rule out the PQWW axion, prompting the development of invisible axion models with higher symmetry-breaking scales.

The PQWW axion model, named after Peccei, Quinn, Weinberg, and Wilczek, is the original implementation of the Peccei-Quinn mechanism proposed to solve the strong CP problem in quantum chromodynamics (QCD) by dynamically relaxing the effective $\bar\theta$ parameter to zero. This model extends the Standard Model with a global chiral $U(1)_{PQ}$ symmetry that is spontaneously broken at the electroweak scale, producing a pseudo-Nambu-Goldstone boson, the axion, as a physical degree of freedom. The PQWW axion acquires couplings to gluons via the QCD anomaly, as well as to photons and fermions, with all interaction strengths fixed by the electroweak scale vacuum expectation value. Experimental and astrophysical constraints have definitively ruled out the viability of the PQWW axion, motivating the development of "invisible" axion models with much higher symmetry-breaking scales (Favitta, 23 Dec 2025, Marsh, 2015).

1. Field Content, Symmetry, and Charge Assignments

The minimal PQWW axion model is realized in a two-Higgs-doublet extension of the Standard Model. The field content is as follows:

• Two complex Higgs doublets: $\Phi_1 \sim (\mathbf{2}, +1/2)$ and $\Phi_2 \sim (\mathbf{2}, +1/2)$, with opposite $U(1)_{PQ}$ charges ($q_{PQ}(\Phi_1)=+1,\, q_{PQ}(\Phi_2)=-1$).
• Standard Model quark and lepton fields, with left-handed doublets ($Q_{L_i},L_{L_i}$) carrying zero $U(1)_{PQ}$ charge. Right-handed up-type quarks $u_{R_i}$ carry $-1$, down-type quarks $d_{R_i}$ and right-handed charged leptons $\ell_{R_i}$ carry $+1$.
• The global $U(1)_{PQ}$ acts axially on the Higgs and chirally on the right-handed fermions, enforcing a specific structure to the Yukawa sector and the scalar potential.

When both Higgs doublets acquire vacuum expectation values (VEVs), $\langle\Phi_i\rangle = v_i/\sqrt{2}$ with $v_{\rm EW}=\sqrt{v_1^2+v_2^2}\approx246$ GeV, the $U(1)_{PQ}$ symmetry is spontaneously broken at the electroweak scale. The physical axion $a(x)$ is the linear combination of the phases of the two Higgs doublets orthogonal to the Goldstone boson eaten by the $Z$ (Favitta, 23 Dec 2025, Marsh, 2015).

2. Lagrangian Structure and Anomaly-Induced Axion Couplings

The PQWW model's renormalizable Lagrangian consists of:

$$\mathcal L = \mathcal L_{\mathrm{kin}} + \mathcal L_{\mathrm{Yukawa}} - V(\Phi_1, \Phi_2) + \mathcal L_{aGG} + \mathcal L_{a\gamma\gamma}$$

After electroweak symmetry breaking and appropriate field rotations, the axion $a(x)$ enters as a pseudo-Nambu-Goldstone field with the kinetic term $(1/2)\partial_\mu a\,\partial^\mu a$. Couplings to QCD and QED field strengths arise via chiral anomalies:

$$\mathcal{L}_{aGG} = \frac{\alpha_s}{8\pi}\frac{a}{f_a}G^a_{\mu\nu} \tilde G^{a\mu\nu}$$

$$\mathcal{L}_{a\gamma\gamma} = -\frac{\alpha_{\rm em}}{8\pi}\frac{C_{a\gamma}}{f_a} a F_{\mu\nu}\tilde F^{\mu\nu}$$

Here $f_a = v_{\rm EW}/N_{\rm DW}$ is the axion decay constant and $N_{\rm DW}=1$ is the domain wall number in the original PQWW construction. The electromagnetic anomaly coefficient is $C_{a\gamma} = (E/N) - 1.92$ with $E/N=0$ in PQWW, so $C_{a\gamma}\simeq -1.92$ (Favitta, 23 Dec 2025, Marsh, 2015).

3. Axion Mass, Potential, and Domain Wall Number

The QCD anomaly generates a periodic axion potential through instanton effects, aligned to eliminate the physical vacuum angle:

$$V(a) \simeq m_\pi^2 f_\pi^2 \left[ 1 - \cos \left( \frac{a}{f_a} \right) \right]$$

Expanding for small $a$ yields an axion mass

$$m_a \approx \frac{m_\pi f_\pi}{f_a} \frac{\sqrt{m_u m_d}}{m_u + m_d} \simeq 6\times10^{-6}\,\text{eV}\,\left( \frac{10^{12}\,\text{GeV}}{f_a} \right)$$

In PQWW, $f_a\sim v_{\rm EW} \approx 250$ GeV, so $m_a$ is at the MeV scale, and the axion has strong interactions with ordinary matter. The domain wall number $N_{\rm DW}$ is unity, reflecting the color anomaly and single-valuedness of the potential (Favitta, 23 Dec 2025, Marsh, 2015).

4. Axion Couplings to Standard Model Fermions and Photons

Yukawa couplings, after symmetry breaking and diagonalization, induce both derivative and pseudoscalar couplings of the axion to Standard Model fermions:

$$\mathcal L_{a \bar{f} f} = \frac{C_{af}}{2 f_a} \partial_\mu a\, \bar{f}\gamma^\mu\gamma_5 f$$

$$C_{ae} \simeq \frac{m_e}{v_{\rm EW}}, \qquad C_{ap} \simeq -0.47, \qquad C_{an}\simeq0.02$$

After Fierz rearrangement, these correspond to pseudoscalar couplings $i g_{a f} a \bar{f}\gamma_5 f$, with $g_{af}\simeq C_{af} m_f / f_a$. Photonic couplings are anomalous and scale as $g_{a\gamma\gamma}= (\alpha/2\pi f_a) C_{a\gamma}$, which for $f_a$ at the electroweak scale gives extremely large values (Favitta, 23 Dec 2025, Marsh, 2015).

5. Phenomenological and Experimental Constraints

The PQWW axion is definitively excluded by a range of laboratory, astrophysical, and cosmological observations:

• Laboratory limits: Beam-dump, rare meson decay, and collider searches exclude axion decay constants $f_a < 10^4$–$10^6$ GeV and thus the PQWW parameter space.
• Stellar cooling: Axion emission from stars (red giants, HB stars) and SN1987A neutrino burst duration constrain $f_a\gtrsim10^8$–$10^9$ GeV, far exceeding the PQWW value.
• Cosmological bounds: Excess axion emission shortens neutrino bursts or alters cosmic evolution; the required coupling suppression is only achieved for $f_a \gg v_{\rm EW}$ (Favitta, 23 Dec 2025, Marsh, 2015, Marsh, 2015).

These bounds collectively exclude the PQWW window $f_a \sim 10^2$–$10^6$ GeV.

6. The Axion "Quality" Problem and Fifth-Force Constraints

Explicit PQ symmetry breaking—e.g., higher-dimensional operators coupling the PQ field to electrons or gluons—induces shifts in the axion potential and vacuum expectation value, endangering the solution to the strong CP problem. Fifth-force and equivalence-principle experiments now set stronger constraints on the scale $\Lambda$ suppressing such operators than the neutron EDM. For the original PQWW realization ($f_a\sim 10^2$ GeV), torsion-balance bounds require $\Lambda/|\sin\Delta| \gtrsim 10^{17}$–$10^{21}$ GeV for axion masses $m_a \sim 10^{-5}$ eV, vastly exceeding possible UV cutoff scales and essentially ruling out low-scale PQWW implementations even for extremely suppressed explicit-breaking terms (Zhang, 2022).

7. Extensions and Evolution: From PQWW to Invisible Axions

The failure of the PQWW axion to survive observational scrutiny led to the development of "invisible" axion models, notably the KSVZ and DFSZ classes, which raise $f_a$ to $10^9$–$10^{12}$ GeV and decouple axion couplings from the electroweak scale. These models remain viable dark matter candidates and are embedded in more complex frameworks such as supersymmetric GUTs (e.g., SO(10) SUSY+PQ) and gauged multi-Higgs models, where the axion is "invisible" and all laboratory, astrophysical, and cosmological bounds can be satisfied (Dias et al., 2021, Baer, 2010).

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References

• (Favitta, 23 Dec 2025) Non-equilibrium Quantum Field Theory and Axion Electrodynamics in curved spacetimes
• (Marsh, 2015) Axion Cosmology
• (Dias et al., 2021) Axion-Neutrino Interplay in a Gauged Two-Higgs-Doublet Model
• (Zhang, 2022) The Axion Quality and Fifth Force
• (Baer, 2010) SO(10) SUSY GUTs with mainly axion cold dark matter: implications for cosmology and colliders