QCD Axion Parameter Space

• QCD Axion Parameter Space is the range of mass and coupling values constrained by the Peccei-Quinn symmetry, addressing the strong CP problem and dark matter relic density.
• Recent advances extend the classic window through mechanisms like recurrent misalignment, multiaxion mixing, and PQ quality constraints, broadening experimental targets.
• Complementary experimental strategies, including haloscopes, gravitational wave detectors, and collider probes, are crucial for testing various QCD axion scenarios.

The QCD axion parameter space comprises the set of mass–coupling relations, cosmological histories, and experimental targets that realize a solution to the strong CP problem through Peccei-Quinn (PQ) symmetry and accommodate the axion as dark matter. The archetypal prediction is a narrow band in mass–coupling space, governed by the topological susceptibility of QCD, astrophysical exclusion, and the relic density supplied via the misalignment mechanism. Recent theoretical advances introduce mechanisms that dramatically broaden the viable parameter space, incorporating UV model-building, cosmological histories, PQ quality constraints, gravitational signatures, direct detection targets, and the possibility of multiple or exotic QCD-axion-like particles.

1. Standard QCD Axion Window: Mass, Couplings, and Relic Abundance

The QCD axion mass is fixed by the relation

$$m_a \simeq 5.7\,\mu\text{eV}\left( \frac{10^{12}\,\text{GeV}}{f_a} \right)$$

where $f_a$ is the axion decay constant. The defining axion–photon coupling is

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

with $E/N$ determined by UV PQ charge assignments (e.g., $E/N=0$ for KSVZ, $=8/3$ for DFSZ) and $1.92$ the model-independent QCD mixing contribution. Axion–nucleon and axion–electron couplings are proportional to $1/f_a$ and are fixed by the axial current structure and PQ charges.

The cosmological relic density from misalignment is

$$\Omega_a h^2 \simeq 0.18\,\theta_i^2\left( \frac{f_a}{10^{12}\,\rm GeV}\right)^{1.19}$$

for a random initial misalignment angle $\theta_i \sim O(1)$, yielding the observed dark matter relic abundance $\Omega_a h^2 \sim 0.12$ at $f_a\sim10^{11}$–$10^{12}$ GeV ($m_a\sim$ few $\mu$eV), for both pre- and post-inflation scenarios, modulo topological defect contributions in the post-inflationary case (Baer, 2015).

Astrophysical and cosmological limits require

$10^9\,\rm GeV\lesssim f_a\lesssim10^{12}\,\rm GeV$

to evade cooling bounds (SN1987A, stellar energy loss) and overclosure, setting $m_a\sim10^{-6}$–$10^{-2}$ eV as the "classic" QCD axion window (Baer, 2015). Direct searches, notably ADMX, are probing the canonical band in $g_{a\gamma\gamma}$–$m_a$ space.

2. Mechanisms Eroding and Extending the Classic Window

Several dynamical effects and UV scenarios generate significant departures from the canonical parameter space.

a. Recurrent Misalignment from FOEWPT

A first-order electroweak phase transition (FOEWPT), coupled to the axion through a non-renormalizable PQ-violating portal, induces an early axion oscillation phase during the FOEWPT epoch (with enhanced effective mass), a subsequent freezing, and then standard late QCD-driven oscillations. This "recurrent misalignment" broadens the cosmologically favored $f_a$ window to

$$10^8\,\mathrm{GeV}\leq f_a\leq 10^{14}\,\mathrm{GeV}$$

for $\theta_i\sim O(1)$, without fine-tuning. The corresponding mass range $m_a\sim 60$ meV $- 0.1\,\mu$eV connects axion dark matter to anticipated gravitational wave signals from the FOEWPT in the mHz–Hz band ($f_{\rm peak}\sim 10^{-3} - 10^{-1}$ Hz), testable by LISA, DECIGO, BBO, and others (Bhandari et al., 7 Jul 2025).

b. Multiaxion Mixing and the Axion Sum Rule

If the PQ sector includes $N$ real pseudoscalars with a single PQ-symmetry and arbitrary mixing, the physical axion eigenstates share the QCD topological susceptibility according to

$$\sum_{i=1}^N m_i^2 f_i^2 = \chi_{\rm QCD}, \qquad m_i^2 f_i^2 = \chi_{\rm QCD}\,g_i, \quad g_i \geq 1, \;\sum_{i=1}^N 1/g_i = 1$$

Every physical axion lies to the right of the canonical QCD band in the $m$–$g_{a\gamma\gamma}$ plane, with the maximal shift for the closest state being $m_i^{\rm max} = \sqrt{N}\,m_a^{\rm QCD}$. Observation of one displaced axion would guarantee the existence and approximate location of other QCD-axion-like states within a calculable region (Gavela et al., 2023).

c. Axion Quality, Planck-Scale Effects, and Discrete Gauge Symmetries

Gravity-induced PQ-breaking via Planck-suppressed operators generically spoils the strong CP solution unless forbidden by extra symmetry. SM-motivated $\mathbb{Z}_4 \times \mathbb{Z}_3$ gauge symmetry permits a "high-quality" QCD axion at

$$10^{11}\,\rm GeV \lesssim f_a \lesssim 2\times10^{12}\,\rm GeV, \qquad 3\times10^{-5}\,\rm eV \lesssim m_a \lesssim 5\times10^{-4}\,\rm eV$$

with a sharply defined red band in $m_a$–$g_{a\gamma\gamma}$, as current/future haloscopes approach or span this region. Quality constraints thus impose an upper bound $F_a\lesssim 2\times10^{12}$ GeV, while cosmological relic abundance sets the lower limit (Sheng et al., 20 Oct 2025). Similar constraints emerge from conformal window strong dynamics and electric-magnetic duality, with PQ-breaking fields in the magnetic theory yielding $$10^{10}\,\rm GeV\lesssim f_a\lesssim10^{12}\,\rm GeV$$ to maintain both DM abundance and PQ quality (Nakagawa et al., 12 Dec 2024).

3. Cosmological and Theoretical Scenarios Broadening the Allowed Parameter Space

a. Isocurvature and Inflationary Dynamics

The classic isocurvature constraint, $H_I/(\pi f_a \theta_i)\lesssim 2.8\times10^{-5}$ (from Planck), ties the maximum inflation scale to the PQ scale. However, with a small quartic PQ self-coupling, a nonminimal gravitational coupling, or heavy-lifting by inflaton-coupled operators, the effective PQ-breaking scale during inflation can be significantly larger than the late-time $f_a$, permitting suppression of isocurvature even at $H_I\sim10^{13}$–$10^{14}$ GeV and $f_a$ up to the Planck scale (Graham et al., 3 Jun 2025, Bao et al., 2022). Parametric resonance and PQ symmetry restoration bounds must also be satisfied to avoid overproduction or domain-wall problems.

b. Late-Time Phase Transitions and Warm Axion Production

In supersymmetric and other scenarios where the PQ phase transition occurs at late times ($T_c\ll f_a$), axions are copiously produced by parametric resonance from oscillating saxions. The observed DM abundance can be realized for $f_a$ as low as $10^8$–$10^{11}$ GeV ($m_a\sim10^{-7}$–$10^{-4}$ eV), with a warm momentum spectrum and observable impact in future 21 cm surveys or rare-decay (FCNC) searches (Harigaya et al., 2019).

c. Dark Photons and Dynamical Axion Depletion

A coupling between the QCD axion and a massless dark photon can trigger tachyonic and parametric resonance production of dark radiation, depleting the axion relic abundance and allowing $f_a$ up to $2\times10^{17}$ GeV for $\theta_i\sim O(1)$, bounded above by $\Delta N_{\rm eff}$ constraints from the CMB. Isocurvature bounds are exponentially relaxed, and axion minicluster formation is enhanced. The apparatus of parameter-space extension is robust to small-angle/parameter variation in the dark photon coupling (Agrawal et al., 2017).

d. Axion Potential Modifications: Hilltop and Anarchy

A light axion with a flattened potential at the origin (e.g., "hilltop" misalignment from a mirror sector with a CP-shifted phase) realizes the correct relic density for $10^9$–$10^{14}$ GeV and $10^{-11}$–$10^{-3}$ eV, occupying regions well beyond the standard misalignment track. Non-adiabatic tracking across the QCD transition alters the predicted abundance, motivating searches at both ultra-low and higher axion masses (Co et al., 17 Jul 2024). Separately, "Anarchy" constructions with soft explicit PQ breaking produce parametrically lighter QCD axions—artificially tuned to preserve $n$EDM viability while occupying previously excluded $m_a$–$g_{a\gamma\gamma}$ space (Elahi et al., 2023).

4. Experimental Coverage and Strategy

a. Haloscopes and Broadband Detectors

The canonical QCD axion band is currently targeted by ADMX, CAPP, and HAYSTAC, sensitive around $m_a\sim1$–$40\,\mu$eV, with QUAX and ORGAN covering up to $10^{-4}$ eV. LC-circuit-based experiments (ABRACADABRA, DMRadio) and nuclear spin precession (CASPEr) extend reach to $m_a \ll \mu$eV. IAXO and its upgrades provide complementary access via helioscopes.

Axion–photon coupling enhancement via the clockwork mechanism can arbitrarily shift experimental targets, allowing, e.g., the same $f_a$ as canonical QCD axion but with $g_{a\gamma\gamma}$ at IAXO or ABRACADABRA sensitivity, accompanied by LHC-testable pseudo-scalar towers (Farina et al., 2016).

b. Axion Magnetic Resonance in Helioscopes

A helical B-field profile (axion magnetic resonance, AMR) allows resonant recovery of coherence at high axion mass, extending the mass reach up to $m_a\sim0.2$ eV and improving $g_{a\gamma\gamma}$ sensitivity by factors of 2–4 in CAST/BabyIAXO/IAXO+ configurations (Seong et al., 20 Aug 2024). This opens direct laboratory access into regions of QCD axion parameter space that overlap with KSVZ/DFSZ targets for the first time in the high-mass regime.

c. Coupling Complementarity: Photon, Nucleon, and Electron Channels

Polarization haloscopes probe axion–nucleon couplings at sensitivity rivaling ADMX’s reach in $g_{a\gamma\gamma}$, enabling a cross-check of the predicted $g_{a\gamma\gamma}/g_{aNN}$ ratio. Confirming both types of signals at consistent frequencies constitutes a model-independent "smoking gun" for a canonical QCD axion (Berlin et al., 2022).

d. Distinctive Gravitational Wave and LHC Signatures

FOEWPT-induced axion cosmologies predict stochastic GW backgrounds at frequencies and amplitudes within LISA, DECIGO, and BBO. Synchronous detection of both axion and GW events coherently points to this sector (Bhandari et al., 7 Jul 2025). Exotic axions (e.g., ultra-light or MeV-scale constructions) entail unique collider signatures, such as $h\to aa\to 4e$ lepton-jet events for low-scale axion models (Liu et al., 2021).

5. Constraints, Exclusion, and Model Discrimination

The QCD axion parameter space is subject to a network of constraints from laboratory bounds (direct detection, rare decays), astrophysical and cosmological limits (SN1987A, stellar cooling, BBN, CMB, $\Delta N_{\rm eff}$, Lyman-$\alpha$ forest), and theoretical consistency (PQ quality, isocurvature, Landau-pole avoidance).

A summary of canonical and broadened parameter regions is encapsulated in the table below.

Mechanism	$f_a$ Range (GeV)	$m_a$ Range (eV)	Key Features/Constraints
Standard Misalignment	$10^9$ – $10^{12}$	$10^{-6}$ – $10^{-2}$	KSVZ/DFSZ/QCD band
FOEWPT Recurrent Misalignment	$10^8$ – $10^{14}$	$10^{-7}$ – $10^{-4}$	Gravitational waves, extended window
High-Quality/Discrete Symmetry	$10^{11}$ – $2\times10^{12}$	$3\times10^{-5}$ – $5\times10^{-4}$	Planck-safe, haloscope-targeted region
Dark Photon Depletion	$10^9$ – $2\times10^{17}$	$10^{-10}$ – $10^{-6}$	$\Delta N_{\rm eff}$, miniclusters, isocurvature
Hilltop Misalignment	$10^9$ – $10^{14}$	$10^{-11}$ – $10^{-3}$	Mirror-tuning, novel oscillation
Late PQ Phase Transition	$10^8$ – $10^{11}$	$10^{-7}$ – $10^{-4}$	Warm DM, 21 cm/rare Kaon decay signals
Anarchic Axion	$10^6$ – $10^9$ (tuned)	$10^{-8}$ – $10^{-6}$	Nonstandard cosmology/anthropic
MeV-scale Axion	$1$ GeV	$10$ MeV	Atomki anomaly, $h\to aa\to4e$ at LHC
Multiaxion Sum Rule	$>10^9$	$>10^{-7}$	Multiple axionlike states, shifted canonical band

Each mechanism's relevant mass, coupling, and cosmological parameters, as well as its exclusion or future reach, are subject to ongoing experimental progress. The intersection of haloscope sensitivity, gravitational wave observatories, collider signals, and low-energy probes will systematically explore, constrain, and potentially resolve the QCD axion parameter space across its theoretical extensions.

6. Outlook: The Future of QCD Axion Parameter Space Exploration

The evolving theoretical landscape—in which new UV completions, cosmological scenarios, and symmetry structures modify the expected mass–coupling relations—now demands a multi-observable strategy. Axion searches must span orders of magnitude beyond the initial predictions, targeting complementary couplings and exploiting emerging detection technologies (LC-circuits, nuclear polarization, GWs, heliscopes with AMR). Concurrent signals in orthogonal channels (photon, nucleon, gravitational) will not only allow discovery but also precise discrimination among QCD axion scenarios, offering the potential to illuminate the global structure of axion parameter space and resolve foundational questions in strong CP, dark matter, and high-scale physics (Bhandari et al., 7 Jul 2025, Gavela et al., 2023, Sheng et al., 20 Oct 2025, Seong et al., 20 Aug 2024, Nakagawa et al., 12 Dec 2024, Baer, 2015).