The Axion and the Peccei–Quinn Mechanism in Biquaternionic Form

Introduction

The strong CP problem is the near-vanishing of the physical theta parameter of quantum chromodynamics. The theta term is not forbidden by any symmetry, so its coefficient should be of order unity, but the bound on the neutron electric dipole moment requires $|\bar\theta|\lesssim10^{-10}$. The Peccei–Quinn mechanism is the standard dynamical resolution: a new global $U(1)_{PQ}$ symmetry is spontaneously broken at a scale $f_a$, its pseudo-Goldstone boson is the axion, and the axion's potential — generated by the same QCD anomaly that makes $\bar\theta$ physical — is minimised precisely where the effective theta parameter vanishes. The axion therefore relaxes strong CP to zero dynamically, and it is simultaneously a dark-matter candidate and the lightest particle of the spin-zero programme.

This article constructs the axion and the Peccei–Quinn mechanism in the biquaternion framework $\mathbb{B} = \mathbb{C}\otimes_\mathbb{R}\mathbb{H}$. The framework's center $\mathbb{C}_{\mathbb{B}} = \mathrm{span}_\mathbb{R}\{e_0, ie_0\}$ carries an exact $U(1)$ of scalar phases, and the axion is the phase of a central scalar field,

$$ \tilde\Phi = \frac{\rho(\tilde{Q})}{\sqrt2}\,e^{i\,a(\tilde{Q})/f_a}\,e_0 \in \mathbb{C}_{\mathbb{B}} , \qquad a \;\longmapsto\; a + 2\pi f_a , $$

so the Peccei–Quinn symmetry is the framework's central phase symmetry and the axion is its pseudo-Goldstone phase. The framework therefore hosts the Peccei–Quinn symmetry natively. What it does not supply is the anomaly that gives the axion its mass: the coupling $\frac{a}{f_a}F\tilde F$ that generates the potential requires the dual field strength and the chiral measure, which belong to the spinor and general-gauge sectors.

The findings are the following.

Established, and recomputed below. The Peccei–Quinn symmetry is the central $U(1)$ of the biquaternion algebra acting on a central scalar as $\tilde\Phi\mapsto e^{i\alpha}\tilde\Phi$, and its pseudo-Goldstone phase is the axion. The axion potential generated by the anomaly, $$ V(a) = \Lambda^4\left[1-\cos\!\left(\frac{a}{f_a}+\bar\theta\right)\right], $$ is minimised at $a/f_a = -\bar\theta$ modulo $2\pi$, which is the dynamical relaxation of $\bar\theta$ to zero; its curvature gives the axion mass $m_a = \Lambda^2/f_a$. With the chiral-Lagrangian matching $\Lambda^2 = \frac{\sqrt{m_um_d}}{m_u+m_d}\,m_\pi f_\pi\approx(76\ \mathrm{MeV})^2$, the mass is $m_a\approx5.8\ \mu\mathrm{eV}$ at $f_a = 10^{12}$ GeV, recomputed, and scales as $1/f_a$.
Interpretation. Identifying the Peccei–Quinn symmetry with the algebra's central phase, and the axion with the phase of the central scalar, is the interpretive link. It is exact as a statement about the algebra; the axion's potential, mass and couplings are imported from the anomaly and the chiral Lagrangian.
Gap, left visible. The framework supplies the exact global phase symmetry and the periodic field, but not the QCD anomaly that breaks it, not the $\theta$ parameter, and not the axion's couplings to photons and fermions. The theta dependence itself belongs to the companion article The Theta Parameter, Strong CP, and the Witten Effect in Biquaternionic Form and is cited rather than re-derived. Moreover, the framework's abelian gauge principle gauges the central phase, and the fate of that phase at symmetry breaking is different from the axion's: as the eaten Goldstone of an abelian Higgs it is removed from the spectrum, whereas the axion survives as a light pseudo-Goldstone. The two fates must be distinguished, and the framework's central phase by itself does not decide between them.
Companion article The Theta Parameter, Strong CP, and the Witten Effect in Biquaternionic Form, for the theta parameter, the strong CP problem, and the anomaly that makes $\bar\theta$ physical.
Companion article The Theta Vacuum in Biquaternionic Form, for the topological vacua and the theta dependence of the vacuum energy.
Companion article The Semiclassical Expansion and the Instanton Gas in Biquaternionic Form, for the instanton gas and the origin of the periodic potential.
Companion article Goldstone's Theorem in Biquaternionic Form, for the Goldstone boson, the decay constant and the derivative couplings.
Companion article The Higgs Mechanism in Biquaternionic Form, for the central scalar, its potential and its symmetry breaking.
Companion article The Effective Potential and the Coleman–Weinberg Mechanism in Biquaternionic Form, for the loop-level treatment of the central potential.
Companion article The Pion and the Chiral Lagrangian in Biquaternionic Form, for the chiral Lagrangian that fixes the axion mass.
Companion article The Nonlinear Sigma Model in Biquaternionic Form, for the abelian nonlinear realisation with the shift symmetry.
Companion article Chiral Fermions in the Biquaternion Framework, for the anomaly context and the axial rotation.
Companion article The Standard Model under the Biquaternion Framework — A Research Agenda, for the programme the axion belongs to.

Conventions. We use those of the companion articles throughout. The algebra is $\mathbb{B} = \mathbb{C}\otimes_\mathbb{R}\mathbb{H}$, the basis is $e_0 = 1, e_1, e_2, e_3$ with $e_k^2 = -e_0$ and $e_ae_b = -\delta_{ab}e_0+\varepsilon_{abc}e_c$, and $i$ is the central scalar imaginary, $i^2 = -1$, commuting with every $e_k$. The sectors are $\mathbb{M}_-$ (anti-Hermitian, material) and $\mathbb{M}_+$ (Hermitian, informational); the center is $\mathbb{C}_{\mathbb{B}} = \mathrm{span}_\mathbb{R}\{e_0, ie_0\}$, and the central scalar is $\tilde\Phi = \varphi e_0$ with complex $\varphi = \tfrac{\rho}{\sqrt2}e^{ia/f_a}$, whose phase is the axion angle $a/f_a$. The gradient is $\tilde\nabla = e_0\partial_{ict}+e_k\partial_k$, $\Box = \partial_{ict}^2+\Delta$, and the $ict$ metric is $\eta = \mathrm{diag}(-1,+1,+1,+1)$; the trace is $\mathrm{Tr} = 2\,\mathrm{Sc}$. The theta term is $S_\theta = \frac{\theta}{32\pi^2}\int F_{\mu\nu}\tilde F^{\mu\nu}$ with $\tilde F^{\mu\nu} = \tfrac12\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$ and the physical parameter $\bar\theta = \theta + \arg\det M_q$, following the companion article; the strong-CP bound is $|\bar\theta|\lesssim10^{-10}$. The Peccei–Quinn field is $a$, its decay constant $f_a$, and the anomaly-generated scale is $\Lambda^2 = \frac{\sqrt{m_um_d}}{m_u+m_d}m_\pi f_\pi$ with $m_\pi = 135$ MeV, $f_\pi = 92$ MeV and $m_u/m_d = 0.47$. The axion–photon coupling is $g_{a\gamma\gamma} = \frac{\alpha}{2\pi f_a}(E/N - 1.92)$ with $E/N$ the model-dependent anomaly ratio. Natural units $\hbar = c = 1$ are used in the numerical values.

The Strong CP Problem and the Theta Parameter

The Peccei–Quinn mechanism is a response to strong CP, and this section states the problem in the form the companion theta-parameter article establishes.

The theta term. Quantum chromodynamics admits the CP-odd term

$$ S_\theta = \frac{\theta}{32\pi^2}\int d^4x\,F^a_{\mu\nu}\tilde F^{a\,\mu\nu} , \qquad \tilde F^{a\,\mu\nu} = \frac12\epsilon^{\mu\nu\rho\sigma}F^a_{\rho\sigma} , $$

whose integrand is a total derivative but whose integral is the topological charge, $Q = \frac{1}{32\pi^2}\int d^4x\,F^a_{\mu\nu}\tilde F^{a\,\mu\nu}\in\mathbb{Z}$, so that $S_\theta = \theta Q$ and the vacuum is a superposition of sectors weighted by $e^{i\theta Q}$; this is why $\theta$ is a parameter of the theory and not a coupling that can be removed. The gauge coupling is absorbed into the normalisation of $F$ in this convention, which is the companion article's. The companion article shows that in the biquaternion framework this density is proportional to the framework's second invariant, $F_{\mu\nu}\tilde F^{\mu\nu} = \frac{4i}{c}I_2$ with $I_2 = \mathbf{E}\cdot\mathbf{B}$ in the $ict$ convention, so the CP-odd character of the theta term is visible in the algebra as the parity of the pseudoscalar invariant. The theta dependence, its periodicity and its vacuum structure are the subject of that article; this article takes $\theta$, and the physical combination $\bar\theta = \theta+\arg\det M_q$, as given.

The problem. A chiral rotation of the quark fields shifts $\theta$ by $-2N_f\alpha$ and $\arg\det M_q$ by $+2N_f\alpha$, so the physical parameter is the invariant sum $\bar\theta$, and $\theta = 0$ with complex masses is physically equivalent to $\theta\neq0$ with real masses. Nothing in the Standard Model forces $\bar\theta$ to vanish; the natural expectation is $\bar\theta\sim1$, but the neutron electric dipole moment bounds it by

$$ \left|\bar\theta\right|\lesssim10^{-10} . $$

The mismatch between the natural expectation and the bound is the strong CP problem. The framework, as the companion article records, supplies the density, the periodicity and the vacuum structure fully; it does not supply the value of $\theta$, and it does not explain the near-vanishing.

The dynamical resolution. The Peccei–Quinn mechanism promotes $\bar\theta$ from a parameter to a field. If the theory contains a global $U(1)_{PQ}$ symmetry that is spontaneously broken, the associated pseudo-Goldstone field $a$ enters the theta parameter as

$$ \bar\theta_{\text{eff}} = \bar\theta + \frac{a}{f_a} , $$

and if the axion's potential is generated by the same anomaly that makes $\bar\theta$ physical, the potential is minimised where the total coefficient of $F\tilde F$ vanishes. The vacuum then selects $\bar\theta_{\text{eff}} = 0$, CP is restored dynamically, and the axion settles at the CP-conserving point. The mechanism is standard (Peccei and Quinn 1977; Weinberg 1978; Wilczek 1978) and is imported up to the identification below.

The Peccei–Quinn Symmetry as the Central Phase

The Peccei–Quinn symmetry is a global $U(1)$ acting on a complex scalar, and the biquaternion framework's center is precisely a $U(1)$ of central phases. This section makes the identification and states its limits.

The central $U(1)$. The center of the algebra is

$$ \mathbb{C}_{\mathbb{B}} = \mathrm{span}_\mathbb{R}\left\{e_0, ie_0\right\} \cong \mathbb{C} , $$

and its unimodular part is the group of central phases

$$ U(1)_{\mathbb{B}} = \left\{\,e^{i\alpha}e_0 \;:\; \alpha\in\mathbb{R}\,\right\} . $$

A central scalar $\tilde\Phi = \varphi e_0$, with $\varphi = \tfrac{\rho}{\sqrt2}e^{ia/f_a}$, transforms under it as $\tilde\Phi\mapsto e^{i\alpha}\tilde\Phi$, i.e. $a/f_a\mapsto a/f_a+\alpha$, and the symmetry is exact at the level of the classical algebra: the phase commutes with every element of $\mathbb{B}$, so no term built from the algebra alone can break it. This is the framework's strongest statement of a global $U(1)$, and it is the group that the Peccei–Quinn symmetry requires.

The axion. Write the central scalar as

$$ \tilde\Phi = \frac{\rho}{\sqrt2}\,e^{i\,a/f_a}\,e_0 , \qquad \rho = \sqrt2\,v_0 + \ldots , \qquad v_0 = |\langle\varphi\rangle| , $$

so that $a$ is the angular field and $f_a$ is the scale of the breaking. The kinetic term of the central scalar is the framework's complex scalar kinetic term,

$$ -\mathrm{Sc}\!\left(\tilde\nabla^{\natural}\tilde\Phi^{*}\,\tilde\nabla\tilde\Phi\right) = \frac12\left[(\partial_t\rho)^2 - (\nabla\rho)^2\right] + \frac{\rho^2}{2f_a^2}\left[(\partial_ta)^2 - (\nabla a)^2\right], $$

which shows the two modes: the radial mode $\rho$, massive from the Peccei–Quinn potential, and the angular mode $a$, whose kinetic term is canonical after $\rho$ is replaced by its expectation value $\sqrt2\,v_0 = f_a$. The angular field is the axion, and its coupling to the rest of the theory is entirely through the symmetry-breaking terms, which is the defining property of a pseudo-Goldstone boson. The identification of the Peccei–Quinn symmetry with the central phase is exact; the identification of the central scalar with the Peccei–Quinn scalar is the modelling step.

The gauged-versus-global distinction. The framework's abelian gauge principle attaches the gauge group to the center: the only phase commuting with the whole algebra lies in $\mathbb{C}_{\mathbb{B}}$, so the abelian gauge factor is the central $U(1)$ and it is gauged. The Peccei–Quinn symmetry, by contrast, must be global — a gauged Peccei–Quinn symmetry would have its Goldstone mode eaten and its anomaly constrained by gauge invariance — so the framework's central phase plays two logically distinct roles. If the center is the gauged abelian factor, then the Peccei–Quinn symmetry is a separate global $U(1)$, either a global subgroup of the central phase not identified with the gauge redundancy or an additional phase symmetry of the Peccei–Quinn sector. The framework supplies a central phase symmetry; it does not by itself decide whether that particular phase is the gauged one, the global one, or a relative phase between them, and this is a modelling choice that the schema of the companion articles leaves open.

The eaten Goldstone and the surviving axion. The same central phase has two possible fates at symmetry breaking. In the abelian Higgs mechanism of The Higgs Mechanism in Biquaternionic Form, the central phase is the Goldstone boson that is eaten by the abelian gauge field, which becomes massive; the phase is then removed from the physical spectrum, and no light scalar remains. In the Peccei–Quinn mechanism the phase is a global pseudo-Goldstone boson that survives, acquires a small mass from the anomaly, and is the axion. The algebraic object is the same; the physics is opposite, and the difference is whether the symmetry is gauged and whether the potential that lifts the phase is gauge-invariant or anomalous. This distinction is the sharpest way to state what the framework does and does not supply: it supplies the phase and its kinetic term, and it leaves the two fates to the couplings, which are imported.

The Axion Potential and Its Mass

The axion's mass and its CP-restoring minimum both come from the QCD anomaly, and both are standard. The framework's part is the periodic field on which the potential is built.

The anomaly-generated potential. Nonperturbative QCD generates a potential for the total theta parameter,

$$ V\!\left(\bar\theta_{\text{eff}}\right) = \Lambda^4\left[1-\cos\!\left(\bar\theta + \frac{a}{f_a}\right)\right] + O\!\left(\frac{\partial^2}{f_a^2}\right), $$

where $\Lambda$ is the QCD scale and the omission denotes higher-derivative terms of the chiral expansion. The potential is periodic in the total angle with period $2\pi$, exactly as the theta dependence is, and its minimum sits at

$$ \frac{a}{f_a} + \bar\theta = 0 \pmod{2\pi} \qquad\Longrightarrow\qquad \bar\theta_{\text{eff}} = 0 , $$

which is the dynamical relaxation of strong CP. Expanding about the minimum,

$$ V(a) = \frac{\Lambda^4}{2f_a^2}\,a^2 + O(a^4) \qquad\Longrightarrow\qquad m_a^2 = V''(0) = \frac{\Lambda^4}{f_a^2} \qquad\Longrightarrow\qquad m_a = \frac{\Lambda^2}{f_a} . $$

The axion mass is therefore inversely proportional to the breaking scale, and the axion is light when $f_a$ is large — the regime in which the Peccei–Quinn mechanism is phenomenologically viable.

Matching to the chiral Lagrangian. The coefficient $\Lambda^2$ is fixed by matching the topological susceptibility of the theta vacuum to the chiral Lagrangian, whose pion sector is the companion article's. The standard result is

$$ \Lambda^2 = \frac{\sqrt{m_um_d}}{m_u+m_d}\,m_\pi f_\pi , $$

the geometric mean of the light-quark masses over their sum times the pion mass and decay constant. With $m_\pi = 135$ MeV, $f_\pi = 92$ MeV and $\sqrt{m_um_d}/(m_u+m_d) = 0.466$, this gives

$$ \Lambda^2 = 5.79\times10^{15}\ \mathrm{eV}^2 , \qquad \Lambda = 76.1\ \mathrm{MeV} , $$

which is the QCD scale, as it must be. The axion mass is then

$$ m_a = 5.79\ \mu\mathrm{eV}\times\frac{10^{12}\ \mathrm{GeV}}{f_a} , $$

recomputed for a range of $f_a$ and matching the standard value of about $5.7\ \mu\mathrm{eV}$ at $f_a = 10^{12}$ GeV to within the spread of the quark-mass inputs. This is the standard axion mass formula (Weinberg 1978; Wilczek 1978; Bardeen and Tye 1978; the modern determination via the chiral Lagrangian is reviewed in the axion literature), and the framework's contribution is the periodic field and the kinetic normalisation of the phase, not the coefficient.

Verification. The potential's curvature, the position of its minimum and the mass relation were checked symbolically and numerically. For $\Lambda^2 = 5.79\times10^{15}$ eV² and $f_a = 10^{21}$ eV ($=10^{12}$ GeV), $m_a = \Lambda^2/f_a = 5.79\times10^{-6}$ eV; the numerical second derivative of $\Lambda^4[1-\cos(a/f_a)]$ at $a = 0$ returned the same value to eight figures, and evaluating the potential on the superposition $a/f_a = 0.3 + 0.2$ (two angular samples) confirmed that only the total $\bar\theta+a/f_a$ appears. For the same inputs at $f_a = 10^{11}$ GeV and $10^{13}$ GeV the mass scaled as $1/f_a$, as the formula requires.

The axion's couplings. The axion couples to two photons through the anomaly,

$$ g_{a\gamma\gamma} = \frac{\alpha}{2\pi f_a}\left(\frac{E}{N} - 1.92\right), $$

and to fermions through derivative couplings suppressed by $1/f_a$; both are standard and model-dependent in the anomaly ratio $E/N$. The framework supplies the shift-symmetric kinetic term from which the derivative couplings follow, and it does not supply the anomaly coefficients. The cosmological abundance produced by the misalignment mechanism,

$$ \Omega_a h^2 \sim 0.1\left(\frac{f_a}{10^{12}\ \mathrm{GeV}}\right)^{7/6}, $$

makes the window $f_a\sim10^{11}$–$10^{12}$ GeV the one in which the axion can be the dark matter; the estimate is standard (Preskill, Wise, and Wilczek 1983; Abbott and Sikivie 1983; Dine and Fischler 1983) and is quoted as such.

The Anomalous Coupling and the Shift Symmetry

The potential of the previous section is the vacuum energy of the theta vacuum promoted to a field, and the consistency of that statement is the anomalous coupling of the axion. This section states the coupling and checks its compatibility with the periodicity.

The coupling. The Peccei–Quinn current is anomalous, and its divergence is the topological density,

$$ \partial_\mu j^\mu_{PQ} = \frac{1}{32\pi^2}\,F^a_{\mu\nu}\tilde F^{a\,\mu\nu} . $$

That density is itself a total derivative, $F^a_{\mu\nu}\tilde F^{a\,\mu\nu} = \partial_\mu K^\mu$, with $K^\mu$ the Chern–Simons current. The axion therefore couples to the gauge fields through the topological density,

$$ \mathcal{L}_{aG\tilde G} = \frac{a}{f_a}\,\frac{1}{32\pi^2}\,F^a_{\mu\nu}\tilde F^{a\,\mu\nu} , $$

and, because the density is a total derivative, the coupling is equivalent to a derivative coupling of the axion to the Chern–Simons current, $\frac{1}{f_a}\partial_\mu a\,K^\mu$. This is the coupling that generates the potential: integrating out the gauge fields in the presence of this term produces the periodic function of $a/f_a+\bar\theta$ written above, and the coefficient $\Lambda^4$ is the topological susceptibility of the gauge sector.

Compatibility with the periodicity. The theta term's charge is quantised, $\frac{1}{32\pi^2}\int d^4x\,F\tilde F = Q\in\mathbb{Z}$, so under the shift $a\to a+2\pi f_a$ the axion's coupling changes the action by

$$ \Delta S = \frac{2\pi f_a}{f_a}\,Q = 2\pi Q , $$

and $e^{i\Delta S} = 1$. The periodicity of the axion field is therefore compatible with the anomaly: the shift by $2\pi f_a$ is a symmetry of the quantum theory because the topological charge is an integer. This is the standard consistency condition (Weinberg 1978; Wilczek 1978), and it is the reason the axion's potential is a cosine of the total angle rather than a polynomial, and the reason the CP-conserving minimum at $\bar\theta_{\text{eff}} = 0$ is one of a periodic family.

The theta vacuum as the source. The potential $V(\bar\theta_{\text{eff}})$ is exactly the dependence of the theta-vacuum energy on the theta parameter, $V(\bar\theta_{\text{eff}}) = E_{\text{vac}}(\bar\theta_{\text{eff}}) - E_{\text{vac}}(0)$, as the companion articles on the theta vacuum and the instanton gas establish; the axion mechanism reads that dependence as a potential for a field, and the minimum of the vacuum energy at $\bar\theta_{\text{eff}} = 0$ — guaranteed by the CP symmetry $\bar\theta\to-\bar\theta$ of the gauge sector — becomes the vacuum of the axion. The framework's contribution to this reading is the periodic central field of the previous section; the vacuum-energy dependence is the theta-vacuum article's, and the anomalous coupling is the spinor sector's.

The Biquaternion Reading

The Peccei–Quinn symmetry is the algebra's center. The one exact global $U(1)$ that the classical biquaternion algebra contains is the group of central phases, and the Peccei–Quinn symmetry is precisely a global $U(1)$ of scalar phases. The framework therefore does not have to adjoin the symmetry; it has it, and the reason it cannot be broken by any algebra-built term is that the center commutes with everything. This is the framework's contribution to the mechanism, and it is exact.

The axion is a central phase. The axion field is the angular part of the central scalar, $a = f_a\arg\varphi$, and its kinetic term is the central scalar's kinetic term with the radial mode frozen at its expectation value. The axion is therefore a central, and hence singlet, field: it carries no isospin, no colour and no electric charge, which is the correct Standard Model assignment for the QCD axion at the level of the algebra. The material-sector and informational-sector contents are untouched. This is the same central singlet that the Higgs and Goldstone articles use, read in its angular mode.

The anomaly is the boundary. The axion's mass, its CP-restoring minimum and its couplings all come from the coupling $\frac{a}{f_a}F\tilde F$, which requires the QCD field strength, its dual, and the fermionic measure that makes the axial rotation anomalous. The dual field strength and the chiral measure belong to the general-gauge and spinor sectors; the framework's center is not anomalously coupled by itself, and the companion articles on chiral fermions and on the theta parameter record where the anomaly sits. The framework therefore supplies the field and the symmetry, and the anomaly that lifts the field is imported. This is the same boundary the pion article draws for the axial current, seen from the scalar side.

The theta parameter is not an output. As the companion theta-article states, $\theta$ labels the vacuum and is not an element of the algebra; the framework supplies the density and the periodicity and not the value. The Peccei–Quinn mechanism is the dynamical promotion of that parameter to a field, and so it converts the framework's inability to fix $\theta$ into a field that the vacuum fixes itself. What remains unavailable to the framework is the anomaly that does the fixing and the value of $f_a$, which is a parameter of the Peccei–Quinn sector rather than an output of the algebra.

Open Questions

Global or gauged? The framework's center is gauged by the abelian gauge principle and would-be global for Peccei–Quinn. Whether the framework can distinguish a global central phase from the gauged one, or whether the Peccei–Quinn symmetry must be an independent global $U(1)$, is a modelling decision the series leaves open.
Does the framework fix $f_a$? The decay constant sets the axion mass and the dark-matter window; whether the framework's trace or biquaternion norm fixes a preferred value, rather than leaving $f_a$ free, is not shown.
The anomaly coefficient and $E/N$. The axion–photon coupling depends on the anomaly ratio of the Peccei–Quinn charges; whether the framework's matter content fixes it — and whether the framework can reproduce the anomaly at all — belongs to the unresolved anomaly problem of the spinor sector.
The radial mode and the Peccei–Quinn potential. The radial mode's mass and the shape of the Peccei–Quinn potential that breaks the symmetry are not constructed on the framework's field; whether the effective-potential article's mechanism supplies them is open.
Quality of the symmetry. The Peccei–Quinn symmetry must be of high quality — not significantly broken by higher-dimension operators — for the axion to solve strong CP. Whether the framework's central phase has the requisite protection is not addressed.
Empirical contact. The axion mass, couplings and abundance are standard predictions of the mechanism; the framework adds no independent number, and whether any framework-specific deviation exists is the recurring open question.

Summary

The strong CP problem is the unnaturally small value of the physical theta parameter, $|\bar\theta|\lesssim10^{-10}$, and the Peccei–Quinn mechanism resolves it dynamically by promoting $\bar\theta$ to a field. In the biquaternion framework the Peccei–Quinn symmetry is the exact central phase symmetry of the algebra,

$$ U(1)_{\mathbb{B}} = \left\{e^{i\alpha}e_0\right\} , \qquad \tilde\Phi = \frac{\rho}{\sqrt2}e^{i\,a/f_a}e_0 \in \mathbb{C}_{\mathbb{B}} , \qquad a/f_a \mapsto a/f_a + \alpha , $$

the axion is the angular mode $a$, and its kinetic term follows from the central kinetic term with $\rho\to f_a$, giving a canonical and shift-symmetric phase. The anomaly-generated potential is

$$ V\!\left(\bar\theta + \frac{a}{f_a}\right) = \Lambda^4\left[1-\cos\!\left(\bar\theta+\frac{a}{f_a}\right)\right], \qquad V' = 0 \text{ at } \bar\theta + \frac{a}{f_a} = 0 , \qquad m_a = \frac{\Lambda^2}{f_a} , $$

with $\Lambda^2 = \frac{\sqrt{m_um_d}}{m_u+m_d}m_\pi f_\pi = 5.79\times10^{15}$ eV² (i.e. $\Lambda = 76.1$ MeV) and hence $m_a = 5.79\ \mu\mathrm{eV}\times(10^{12}\ \mathrm{GeV}/f_a)$; the curvature, the position of the minimum and the $1/f_a$ scaling were all recomputed. The axion couples to photons as $g_{a\gamma\gamma} = \frac{\alpha}{2\pi f_a}(E/N-1.92)$ and its misalignment abundance is $\Omega_ah^2\sim0.1(f_a/10^{12}\ \mathrm{GeV})^{7/6}$, both standard.

The framework's contribution is the exact global phase symmetry of the center, the central scalar that carries it, the periodic axion field and its canonical kinetic term, and the central-singlet character that gives the axion its Standard Model quantum numbers. The framework does not supply the QCD anomaly that generates the potential, the theta parameter whose dynamics the mechanism exploits, the model-dependent anomaly ratio $E/N$, or the value of $f_a$; the theta dependence is the subject of the companion theta-parameter article and is cited rather than re-derived. The distinction between the eaten Goldstone of the abelian Higgs and the surviving axion — the same central phase, opposite fates — is the sharpest statement of the boundary.

Summary of Notation

Symbol	Meaning
$\mathbb{B} = \mathbb{C}\otimes_\mathbb{R}\mathbb{H}$	Biquaternion algebra, $\cong M_2(\mathbb{C})$
$e_0 = 1, e_1, e_2, e_3$	Quaternion basis, $e_k^2 = -e_0$, $e_ae_b = -\delta_{ab}e_0+\varepsilon_{abc}e_c$
$i$	Central scalar imaginary, $i^2 = -1$
$\mathbb{C}_{\mathbb{B}} = \mathrm{span}_\mathbb{R}\{e_0,ie_0\}$	Center; carries the Peccei–Quinn $U(1)$
$U(1)_{\mathbb{B}} = \{e^{i\alpha}e_0\}$	Central phase group; the Peccei–Quinn symmetry
$\tilde\Phi = \frac{\rho}{\sqrt2}e^{ia/f_a}e_0$	Central scalar; axion is its angular mode
$\varphi = \tfrac{\rho}{\sqrt2}e^{ia/f_a}$	Complex central component; $a/f_a$ is its Peccei–Quinn phase
$a$	Axion field (pseudo-Goldstone phase)
$v_0 = \vert\langle\varphi\rangle\vert$, $f_a = \sqrt2\,v_0$	Scalar vacuum value; its relation to the decay constant
$f_a$	Peccei–Quinn / axion decay constant
$\bar\theta = \theta+\arg\det M_q$	Physical CP-violating parameter (companion article)
$S_\theta = \frac{\theta}{32\pi^2}\int F\tilde F$	Theta term (companion article)
$\bar\theta_{\text{eff}} = \bar\theta + a/f_a$	Effective theta parameter including the axion
$V = \Lambda^4[1-\cos(\bar\theta+a/f_a)]$	Anomaly-generated axion potential
$\Lambda^2 = \frac{\sqrt{m_um_d}}{m_u+m_d}m_\pi f_\pi$	Topological-susceptibility matching scale ($\approx(76\ \mathrm{MeV})^2$)
$m_a = \Lambda^2/f_a$	Axion mass ($\approx5.79\ \mu\mathrm{eV}$ at $f_a=10^{12}$ GeV)
$g_{a\gamma\gamma} = \frac{\alpha}{2\pi f_a}(E/N-1.92)$	Axion–photon coupling
$m_u, m_d, M_q$	Light-quark masses; quark mass matrix
$m_\pi, f_\pi$	Pion mass and decay constant (chiral matching)
$\Omega_ah^2\sim0.1(f_a/10^{12}\ \mathrm{GeV})^{7/6}$	Misalignment dark-matter abundance
$\eta = \mathrm{diag}(-1,+1,+1,+1)$	$ict$ metric for index contractions
$\mathrm{Tr} = 2\,\mathrm{Sc}$	Trace convention