Naturally quality-safe GeV axion with charm coupling

Abstract: The GeV-scale QCD axion -- where Peccei-Quinn (PQ) symmetry is broken by the QCD condensate at $f_a\sim\mathcal{O}(1)$~GeV -- faces a structural isospin problem: the PQ spurion coupling to light quarks ($u,d,s$) breaks $\mathrm{SU}(2)$ isospin, generating an unacceptable $\sim 15\%$ $π<sup>0$--$π<sup>\pm$ mass splitting. We show that coupling the PQ scalar to the charm quark instead eliminates this violation entirely, and lowering $m_φ\sim 3$--$4$~MeV makes the charm Yukawa $κ<em>c\propto m</em>φ$ perturbative ($κ<em>c&lt;1$). The resulting $f_a\sim\text{GeV}\ll M</em>{\rm Pl}$ solves the axion quality problem: even the lowest-dimension $d=6$ Planck-suppressed operator gives $m_{\rm PQ}/m_a\sim 10<sup>{-14}$, without any additional symmetry. The model predicts a distinctive {\it negative} $ΔN_{\rm eff}\sim -0.1$ for $m_φ\sim 3$~MeV, testable by CMB-S4. The $B\to Kσ$ penguin predicts $\mathrm{BR}\sim 2\times 10<sup>{-5}$, consistent with the Belle~II evidence for $B^+\to K<sup>+ν\barν$ at $(2.3\pm0.7)\times 10<sup>{-5}$~\cite{BelleII:2024knv}. All ten classes of experimental, astrophysical, and cosmological constraints are satisfied in the viable window $m_φ\sim 3$--$4$~MeV, with the $B_s$ mixing constraint pending a dedicated lattice calculation.

• The paper introduces a GeV-scale axion model that uses charm coupling to eliminate leading isospin-violating operators and address strong CP issues.
• It employs QCD condensate dynamics and the SVZ sum rule to set precise parameters, ensuring perturbativity and compatibility with meson phenomenology.
• The model naturally suppresses dangerous Planck-scale PQ-breaking effects, yielding predictions testable through collider, astrophysical, and cosmological probes.

Naturally Quality-Safe GeV Axion with Charm Coupling: Technical Analysis

Overview and Motivation

The paper "Naturally quality-safe GeV axion with charm coupling" (2607.12956) proposes a novel solution to the strong CP problem in quantum chromodynamics (QCD) using a GeV-scale axion model, wherein Peccei-Quinn (PQ) symmetry breaking occurs at a scale $f_a \sim \mathcal{O}(1)$ GeV. This construction exploits QCD condensate dynamics to generate the axion and fundamentally differs from the standard 'invisible axion' paradigm ($f_a \sim 10^{9\text{--}12}$ GeV), which is severely decoupled and presents detection challenges. The key advantage of the proposed framework is its resolution of both the axion quality problem and the structural isospin violation that plague earlier low-scale axion models.

Structural Isospin Problem: Diagnosis and Resolution

The Murayama model (Murayama, 5 Jan 2026) demonstrates that PQ symmetry broken by the QCD condensate, with direct coupling to light quarks ($u, d, s$), causes a problematic explicit breaking of $\mathrm{SU}(2)_V$ isospin. Specifically, the PQ spurion $I_{\rm PQ}$ induces a leading-order chiral perturbation theory (χPT) operator $\mathcal{O}_1^{\rm PQ}$, yielding a $-15\%$ $\pi^0$--$\pi^\pm$ mass splitting and $\mathcal{O}(1)$ distortions in pion scattering amplitudes—both irreconcilable with experimental data.

The resolution advanced here is to couple the PQ scalar exclusively to the charm quark ($f_a \sim 10^{9\text{--}12}$0). Since the heavy-quark spurion does not enter the χPT Lagrangian for light mesons, this choice eliminates all isospin-violating operators at leading order. The PQ symmetry and charge assignments forbid bare charm mass terms while preserving Standard Model Yukawas for light quarks, thereby avoiding tree-level FCNCs and reconciling model building with observed low-energy QCD phenomenology.

Parameter Space and Model Construction

The self-consistency of the model relies on the SVZ sum rule for the charm condensate, relating the charm Yukawa $f_a \sim 10^{9\text{--}12}$1 to the PQ scalar mass $f_a \sim 10^{9\text{--}12}$2:

$f_a \sim 10^{9\text{--}12}$3

where $f_a \sim 10^{9\text{--}12}$4 GeV$f_a \sim 10^{9\text{--}12}$5. For $f_a \sim 10^{9\text{--}12}$6--$f_a \sim 10^{9\text{--}12}$7 MeV, $f_a \sim 10^{9\text{--}12}$8--$f_a \sim 10^{9\text{--}12}$9 is perturbative, $u, d, s$0--$u, d, s$1 GeV, and $u, d, s$2--$u, d, s$3 MeV. The axion-photon coupling $u, d, s$4 is substantial, $u, d, s$5 GeV$u, d, s$6, a direct consequence of the model’s KSVZ-like anomaly structure.

Axion Quality Problem: Natural Suppression

A central result is the suppression of Planck-suppressed PQ-breaking operators. For $u, d, s$7 GeV, even the dimension-six operator, $u, d, s$8, induces a shift $u, d, s$9, obviating the need for additional discrete symmetries (contrary to standard models requiring $\mathrm{SU}(2)_V$0 for safety). The quality problem is thus solved via scale selection alone.

Experimental and Cosmological Constraints

All ten major experimental, astrophysical, and cosmological constraints are satisfied for $\mathrm{SU}(2)_V$1--$\mathrm{SU}(2)_V$2 MeV. Key points include:

• Perturbativity: Ensured for $\mathrm{SU}(2)_V$3.
• FCNC: Tree-level forbidden, loop-induced and GIM-suppressed in $\mathrm{SU}(2)_V$4 decays ($\mathrm{SU}(2)_V$5), while $\mathrm{SU}(2)_V$6 is within current Belle II bounds.
• Supernova and Stellar Cooling: The axion is trapped in SN1987A cores due to strong nucleon coupling; stellar cooling constraints are abated by heavy axion mass.
• Cosmology: For $\mathrm{SU}(2)_V$7, axion decays before neutrino decoupling, resulting in $\mathrm{SU}(2)_V$8; near $\mathrm{SU}(2)_V$9 MeV, $I_{\rm PQ}$0, testable by CMB-S4.

  Figure 1: Total invisible branching ratio $I_{\rm PQ}$1 versus $I_{\rm PQ}$2 compared with Belle II measurement; viable window and perturbativity bounds are highlighted.

The predicted $I_{\rm PQ}$3 branching ratio at $I_{\rm PQ}$4 MeV ($I_{\rm PQ}$5) aligns with Belle II observation for $I_{\rm PQ}$6, accounting for the observed excess over SM expectations.

Phenomenological Signatures and Future Probes

The axion's strong photon coupling opens various detection channels:

• Neutral Meson Decays: $I_{\rm PQ}$7, $I_{\rm PQ}$8, searchable in fixed-target and collider setups (Belle II, BESIII, SHiP).
• Missing Energy Signals: $I_{\rm PQ}$9 yields a distinctive long-lived scalar escaping the detector, with decay length $\mathcal{O}_1^{\rm PQ}$0 tens to hundreds of meters.

Potential tension with NA64-e exclusions for axion-photon coupling is addressed; full interpretation requires dedicated simulations incorporating strong axion-nucleon interactions.

Theoretical Implications and Small Quartic Prediction

The model naturally predicts an anomalously small quartic coupling $\mathcal{O}_1^{\rm PQ}$1, far below the Coleman-Weinberg loop correction from the charm sector—this points toward a protecting symmetry, e.g., identifying the PQ scalar as a PNGB from a higher-scale $\mathcal{O}_1^{\rm PQ}$2 breaking.

Model extensions to the $\mathcal{O}_1^{\rm PQ}$3 and $\mathcal{O}_1^{\rm PQ}$4 quark are excluded by BBN and perturbativity, ensuring charm-specific phenomenology.

Outlook and Future Developments

The charm-coupled GeV axion paradigm offers a minimal, technically natural solution to the strong CP and quality problems. Characteristic signatures include:

• A negative $\mathcal{O}_1^{\rm PQ}$5, uniquely testable with future CMB-S4 sensitivity.
• Collider signatures through $\mathcal{O}_1^{\rm PQ}$6 and rare meson decay searches.

Future work will focus on:

• Precise Boltzmann calculations of cosmological signals.
• PNGB model completions for the small $\mathcal{O}_1^{\rm PQ}$7 issue.
• Lattice QCD computations for $\mathcal{O}_1^{\rm PQ}$8 mixing matrix elements.
• Continued Belle II analyses with kinematic discrimination between $\mathcal{O}_1^{\rm PQ}$9 and neutrino channels.

Conclusion

The GeV-scale charm-coupled axion model achieves a technically natural solution for both the CP and quality problems, with a viable parameter window tightly constrained yet phenomenologically rich. Its cosmological and collider signatures are accessible to current and upcoming experiments, and NP quartic predictions are amenable to future UV completions.

Figure 1: Total invisible branching ratio $-15\%$0 as a function of $-15\%$1, with Belle II measurement and theory constraints, indicating the viable window for $-15\%$2--$-15\%$3 MeV.