Axion–SM Fermion Couplings: Theory & Applications

• Axion–SM fermion couplings are derivative and pseudoscalar interactions linking axions with quarks and leptons, fundamental in addressing the strong CP problem.
• They manifest in both flavor-diagonal and off-diagonal forms, with specific coefficients determined by PQ charges and model-dependent Higgs dynamics.
• Their experimental signatures—from meson decays to EDM oscillations and collider production—provide stringent tests of axion and ALP theories.

Axion–Standard Model (SM) fermion couplings constitute the central low-energy signature of Peccei-Quinn (PQ) solutions to the strong CP problem, generic axion-like particle (ALP) extensions, and related pseudo-Nambu-Goldstone bosons. These couplings, typically of derivative (shift-symmetric) form, characterize the interactions of axions with quarks and leptons, control decay and production rates, define the structure of axion-induced anomalies, and set the experimental and cosmological reach of axion searches. The structure, scaling, and anomaly properties of these couplings depend sensitively on model realization, flavor structure, ultraviolet (UV) completion, and the environment (e.g., superfluids, curved spacetime).

1. Structural Forms of Axion–Fermion Couplings

The general low-energy effective Lagrangian for axion–fermion couplings can be written in two equivalent bases: a derivative (shift-symmetric) basis and a pseudoscalar (Yukawa-like) basis. For a Dirac field $\psi$ (quark or lepton), the interaction takes the form

$$\mathcal{L}_{\rm int} = \frac{\partial_\mu a}{f_a} \bar{\psi} \gamma^\mu \gamma^5 \psi \,,$$

where $a(x)$ is the axion field and $f_a$ is the axion decay constant. Upon integrating by parts and using the Dirac equation, this is equivalent (for massive fermions) to a pseudoscalar coupling: $$\mathcal{L}_{\rm int} = i\,\frac{m_\psi}{f_a}\, a\,\bar{\psi}\gamma^5\psi \,.$$ In variant DFSZ and general ALP models, the coefficient can be made explicit via model-dependent parameters: for each fermion $f$,

$$\mathcal{L}_{af} = \frac{C_f}{2f_a}\,(\partial_\mu a)\,\bar{f}\gamma^\mu\gamma^5 f \quad\to\quad i\,\frac{C_f m_f}{f_a}\,a\,\bar{f}\gamma^5 f \,,$$

with $C_f$ proportional to the PQ charge assignment and Higgs structure (Sun et al., 2020, Garcia et al., 2023).

For flavor-off-diagonal (flavor-violating) couplings, the most general interaction is

$$\mathcal{L}_{\rm flav} = \frac{1}{2f_a} (\partial_\mu a) \bar{\psi}_i \gamma^\mu \big(C_{ij}^V + C_{ij}^A \gamma_5 \big) \psi_j \,,$$

where $C_{ij}^{V,A}$ encapsulate mixing angles and PQ charge misalignments (Ziegler, 2023, Bonnefoy et al., 2020). In many ALP extensions, these couplings are significant.

2. Anomalous Ward Identities and Quantum Effects

Classically, the coupling $$\frac{\partial_\mu a}{f_a} \bar{\psi} \gamma^\mu \gamma^5 \psi$$ is associated with a chiral symmetry whose conserved current is $J_5^\mu = \bar{\psi}\gamma^\mu\gamma^5 \psi$. In the quantum theory, this current develops anomalous divergences in the presence of gauge or gravitational backgrounds. The full anomalous Ward identity is

$$\begin{aligned} \left< \nabla_\mu J_5^\mu \right> = & - \frac{1}{12\pi^2 f_a}\, \Box^2 a - \frac{1}{12\pi^2 f_a}\, \nabla_\mu \big[ G^{\mu\nu} \partial_\nu a \big] + \frac{1}{3\pi^2}\, \nabla_\mu \left[ \frac{\partial^\mu a}{f_a} \left(\frac{\partial a}{f_a} \right)^2 \right] \ & - \frac{1}{16\pi^2} \epsilon^{\mu\nu\alpha\beta} F_{\mu\nu} F_{\alpha\beta} + \frac{1}{384\pi^2} \epsilon^{\mu\nu\alpha\beta} R^{\rho\sigma}{}_{\mu\nu} R_{\rho\sigma\alpha\beta} \,, \end{aligned}$$

where the axion-dependent terms are total derivatives related to the derivative coupling (“spurious anomalies”), while the gauge and gravitational terms are the genuine anomalies (Adshead et al., 2021). These “spurious” terms can be removed by local counterterms and originate from the nontrivial Jacobian in the path integral measure under chiral rotations (Fujikawa method).

3. Flavor Structure: Diagonal and Off-diagonal Couplings

The form and strength of axion–fermion couplings depend crucially on the alignment (or misalignment) between PQ charges and the SM Yukawa matrices:

• Flavor-diagonal couplings occur if PQ charges align with the Yukawa matrices. In KSVZ models, $C_f = 0$ for all SM fermions; in DFSZ models, universal diagonal couplings arise, with explicit parameter dependence on Higgs sector vevs and PQ charges (Sun et al., 2020, Garcia et al., 2023).
• Flavor-violating/off-diagonal couplings require misalignment: $[{\rm PQ}, YY^\dagger] \neq 0$. Realistic Froggatt–Nielsen, variant DFSZ, or bulk Higgs (in 5D) models produce off-diagonal $C_{ij}$, typically proportional to CKM mixing or model-dependent parameters (Ziegler, 2023, Bonnefoy et al., 2020). The off-diagonal effective scale $F_{ij} = 2f_a / C_{ij}$ can be as low as $10^{11}$ GeV, within reach of precision flavor experiments.

Benchmarks for $C_{ij}$ in various models are: | Transition | $C_{ij}$ Scaling | Example Value | |-----------------|-------------------------------------|---------------| | $s \to d$ | $V_{us}$, $U(1)_F$ | $\sim 0.22$ | | $b \to d$ | $V_{td}$, $U(1)_F$ | $8\times10^{-3}$| | $\mu \to e$ | PQ charge difference | $O(0.1)$ |

4. Physical Consequences: Production, Decay, and Detection

The phenomenology of axion–SM fermion couplings is determined by the explicit interaction strengths, decay widths, and induced observables:

• Axion decay widths: For the coupling $g_{af} = C_f m_f / f_a$, the partial width for $a\to f\bar{f}$ is

$$\Gamma(a \to f\bar{f}) = \frac{N_c^f\, m_a\, m_f^2}{8\pi\, f_a^2} |C_f|^2 \sqrt{1 - \frac{4m_f^2}{m_a^2}}$$

Once $m_a > 2m_t$, $a\to t\bar{t}$ dominates (Chigusa et al., 26 Feb 2025).

• Flavor-changing decays and off-diagonal channels: Meson decays such as $K \to \pi a$, $B \to K a$, and lepton $f_i \to f_j a$ probe $C_{ij}$ with sensitivities determined by $F_{ij}$. NA62: $F_{sd} > 10^{12}$ GeV, Mu3e/MEG II: $F_{\mu e} > 10^{10}$–$10^{11}$ GeV; current and future flavor factories aim to improve these bounds (Ziegler, 2023).
• Collider production: At high energies, associated production and decay channels such as $\mu^+\mu^- \to t\bar{t} a$ (dominant for heavy ALPs with large $c_t$) provide principal discovery channels for $m_a > 2 m_t$, with sensitivity up to $|c_t/f_a| \sim 6$ TeV$^{-1}$ at $\sqrt{s}=5$ TeV (Chigusa et al., 26 Feb 2025).
• Axion-induced electric and magnetic dipole effects: In the nonrelativistic limit, the derivative axion–fermion coupling induces
  • “axion wind” spin-precession effects via $\nabla a \cdot \vec{\sigma}$,
  • oscillating electric dipole moments $d_{a}(t) \propto (C_\psi e/2m_\psi f_a)a(t)$ for charged fermions, unscreened by Schiff’s theorem, with direct application to EDM search strategies (Smith, 2023, Luzio et al., 2023).

For ALP dark matter, experimental sensitivities cover broad ground: oscillating EDMs are most relevant for fast oscillation regimes ($m_a \gtrsim T^{-1}$ for an experiment of duration $T$), while NMR/comagnetometer experiments probe the “axion wind” (Luzio et al., 2023).

5. Theoretical Constraints: Renormalization, Anomalies, and Cosmology

• Renormalization effects: RG running between UV and low scales induces $O(10\%)$ nonuniversality in $c_q$, $c_\ell$; in universal benchmarks, $c_u \simeq 0.85$, $c_d \simeq 0.96$, $c_e \simeq 0.98$ at $\mu=2$ GeV with $\Lambda=1$ TeV (Garcia et al., 2023).
• Chiral rotations and anomaly distributions: Gluonic ALP couplings can be exchanged for shifts in quark axion couplings via chiral field redefinitions; anomaly matching preserves the low-energy signature (Garcia et al., 2023, Sun et al., 2020).
• Cosmological bounds: Axion–fermion couplings contributing to thermal equilibrium in the early universe are constrained by $\Delta N_{\rm eff}$ during BBN and CMB epochs. For $f_a/C_i < \Lambda_i^{\rm FO}$ (freeze-out above EW scale), these translate to lower bounds on $f_a/C_i$ up to $10^{13}$ GeV for $t$ quark and $10^{11}$ GeV for $\tau$, $b$, $c$ (Green et al., 2021). Table of freeze-out bounds:

| flavor $i$ | $m_i$ [GeV] | $\Lambda_i^{\rm FO}$ [GeV] | |---|---|---| | $e$ | $5.1\times10^{-4}$ | $5.0\times10^7$ | | $\mu$ | $1.06\times10^{-1}$ | $1.2\times10^{10}$ | | $\tau$ | $1.78$ | $2.0\times10^{11}$ | | $t$ | $172$ | $3.6\times10^{13}$ |

• Stellar and supernova constraints: For light axions, stellar cooling via bremsstrahlung sets $f_a/C_e \gtrsim 1.2\times10^{10}$ GeV; supernova neutrino durations constrain $f_a/C_\mu \gtrsim 3\times10^8$ GeV (Green et al., 2021, Ziegler, 2023). Cosmological constraints, however, are sometimes stronger for heavy flavors.

6. Model Realizations: UV Origins and Magnitude Hierarchies

• DFSZ models: Tree-level couplings, set by PQ charge/higgs content. $C_u$, $C_d$, and $C_e$ depend on $\tan\beta$ and Higgs doublet content; universal, no tree-level FCNC unless additional Higgs doublets with nonuniversal PQ charges are present (Sun et al., 2020).
• KSVZ models: SM fermions are PQ neutral, so tree-level axion–fermion couplings vanish; only anomaly-induced couplings from heavy fermion triangle diagrams are present, yielding

$$g_{aff} \simeq \frac{m_f}{f_a}\times \frac{\alpha^2}{4\pi^2}\times\log\left(\frac{M_{\rm heavy}}{m_f}\right)$$

and thus typically suppressed by $\sim 10^{-3}$–$10^{-5}$ relative to DFSZ (Nomura et al., 2020).

• Sterile neutrino/majoron (composite axion) sector: Four-fermion-induced PQ breaking yields a composite axion with $f_a \sim 246$ GeV but $g_{a\gamma},g_a^f \ll 1$, well below laboratory and astrophysical bounds due to form-factor and loop suppression (Xue, 2020).
• Flavored/pseudo-Goldstone ALPs: 5D/warped models, Froggatt–Nielsen, or bulk Higgs scenarios realize flavor-off-diagonal $C_{ij}$ at tree level. Predicted scales $F_{ij}$ are within reach of existing and planned flavor factories (Bonnefoy et al., 2020, Ziegler, 2023).

7. Environmental Effects: Superfluids and Curved Spacetime

In media where fermion number is not conserved (e.g., superfluids), new axion–fermion couplings proportional to emergent Majorana masses arise: $$\mathcal{L}_a^{\rm sf} = -i \frac{(b + c) \Delta^*}{F} a e e + h.c.$$ with $\Delta$ the pairing gap; these terms are suppressed by $\Delta/F$ compared to vacuum couplings (Wilczek, 2014). Analogous couplings exist for nucleons in neutron stars.

Gravitational and cosmological backgrounds induce covariant corrections to the anomalous divergence, including Einstein tensor and curvature couplings, leading to higher-derivative operators in the effective axion action (Adshead et al., 2021).

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In conclusion, the structure, scaling, and phenomenological consequences of axion–SM fermion couplings are determined by the PQ charge assignments, Higgs sector content, and possible flavor violating dynamics, as well as loop anomalies and environmental effects. These couplings are central to the experimental, cosmological, and astrophysical search strategies for axions and ALPs, and provide a window into UV dynamics, flavor structure, and the mechanism of PQ symmetry breaking.