Axion-Photon-Dark Photon Coupling

• Axion–photon–dark photon coupling is defined as the interaction among axions, SM photons, and hypothetical dark photons via higher-dimensional operators, extending dark sector phenomenology.
• It enables novel experimental probes such as resonant cavity, NMR, and collider searches, influencing dark matter production and astrophysical signatures.
• The coupling modifies cosmological and laboratory constraints, with implications for dark photon dark matter, X-ray anomalies, and potentially the muon (g–2) anomaly.

Axion–photon–dark photon coupling refers to a class of interactions connecting three light bosonic fields: the axion (or a generic axion-like particle, ALP), the Standard Model (SM) photon, and the hypothetical dark photon associated with an extra, often hidden, U(1) gauge symmetry. Such couplings extend both the phenomenology and the discovery potential of dark sector physics, enabling novel mechanisms for laboratory searches, cosmological production, astrophysical signatures, and connections to outstanding particle physics anomalies.

1. Effective Lagrangians and Origin of the Coupling

The minimal ingredients comprise kinetic terms for all three fields, kinetic mixing between the photon and dark photon, the standard axion–photon coupling, and, crucially, higher-dimensional operators directly coupling the axion simultaneously to both photons: $$\begin{aligned} \mathcal{L} &\supset -\frac{1}{4} F_{\mu\nu} F^{\mu\nu} -\frac{1}{4} F'_{\mu\nu} F'^{\mu\nu} -\frac{\epsilon}{2} F_{\mu\nu} F'^{\mu\nu} \ &\quad -\frac{g_{a\gamma\gamma}}{4} a F_{\mu\nu} \tilde{F}^{\mu\nu} -\frac{g_{a\gamma'\gamma'}}{4} a F'_{\mu\nu} \tilde{F}'^{\mu\nu} -\frac{g_{a\gamma\gamma'}}{2} a F_{\mu\nu} \tilde{F}'^{\mu\nu}, \end{aligned}$$ where $a$ is the axion, $F_{\mu\nu}$ and $F'_{\mu\nu}$ are the electromagnetic and dark photon field strength tensors, and tildes denote dual tensors.

The portal coupling $g_{a\gamma\gamma'}$ (and $g_{a\gamma'\gamma'}$), often called the "dark axion portal" (Kaneta et al., 2016), is generated via UV-complete models involving heavy fermions charged under both the PQ (axion) and dark $U(1)$ symmetries. These triangle anomaly diagrams generate effective couplings, e.g.,

$$G_{a\gamma\gamma'} \simeq \frac{ee'}{8\pi^2} \frac{PQ_\Phi}{f_a} [2N_C D_\psi Q_\psi] + G_{a\gamma\gamma},$$

where $e,e'$ denote electromagnetic and dark gauge couplings, $Q_\psi, D_\psi$ are electromagnetic and dark charges, $N_C$ is a color factor, and $f_a$ is the axion decay constant (Kaneta et al., 2016, Kaneta et al., 2017). These couplings are not simple products of the vector and axion portals but have intrinsic structure. Importantly, such couplings remain non-zero even if either the standard axion–photon or kinetic mixing is suppressed.

2. Mixing Mechanisms and Three-Way Oscillations

The axion–photon–dark photon system exhibits a rich mixing structure when embedded in environments with background electromagnetic and/or dark sector fields. The equations of motion in the presence of external fields become coupled and, after Fourier decomposition, the system can be modeled as a five-state Schrödinger-like equation for $(A_+, A_\times, A'_+, A'_\times, a)$: $(i \partial_t - k) \Psi_k(t) + M\, \Psi_k(t) = 0,$ with $M$ the mixing matrix containing medium, external field, and axion coupling effects (Ejlli, 2016). Perturbative solutions indicate that transitions between photon, dark photon, and axion are possible, governed by the respective coupling constants and the presence of both ordinary and dark magnetic fields.

Resonant conversion between photons and dark photons mediated by the axion can occur when the in-medium photon effective mass matches the shifted mass eigenvalues of the mixed system, often leading to sharp and frequency-dependent transition probabilities (Choi et al., 2019). These effects can be described by generalized Landau–Zener–type expressions, with conversion probabilities enhanced in the presence of strong external magnetic fields, either visible or dark.

3. Cosmological and Astrophysical Consequences

Mechanisms involving axion-photon-dark photon couplings can dominate several cosmological processes:

• Dark Photon Dark Matter Production: Oscillating axion fields in the early universe can efficiently transfer energy to dark photons via tachyonic instability (Agrawal et al., 2018, Co et al., 2018, Zhang et al., 28 Jul 2025). The relevant instability condition for the Fourier mode $k$ of the dark photon is:

$$\ddot{A}_{k,\pm} + H \dot{A}_{k,\pm} + \left[ m_{\gamma'}^2 + \frac{k^2}{a^2} \mp \frac{k}{a}\left(\frac{\beta \dot\phi}{f_a}\right) \right] A_{k,\pm} = 0,$$

where $\beta$ is the dimensionless coupling, and the last term can make one helicity tachyonic. For $m_{\gamma'}/m_a = O(10^{-3}\text{–}1)$, energy transfer is extremely efficient, producing dark photons that can constitute the dark matter (Agrawal et al., 2018).

• Freeze-In and Two-Component DM: In dark axion portal scenarios, out-of-equilibrium production channels such as $fa \to f \gamma'$ (dark Primakoff process) provide a nonthermal source of dark photon dark matter, supplementing the axion and resolving relic density deficits in regions where axions alone cannot saturate $\Omega_\text{DM}$ (Kaneta et al., 2017).
• Decay Signatures and X-Ray Anomalies: The new coupling $g_{a\gamma\gamma'}$ enables dark photon decays $\gamma' \to a \gamma$ with a width $$\Gamma(\gamma' \to a \gamma) = (g_{a\gamma\gamma'}^2 / 96\pi) m_{\gamma'}^3(1 - m_a^2/m_{\gamma'}^2)^3$$ (Kaneta et al., 2016, Kaneta et al., 2017). For keV-scale dark photons, these channels can explain observed X-ray lines, such as the $3.5$ keV excess.
• Spectral Distortions and CMB Constraints: Resonant or off-resonant conversion between photons, dark photons, and axions in the early universe through $\phi F \tilde{F}_D$-type terms can give rise to unique spectral distortions in the cosmic microwave background, subject to strong constraints from FIRAS and future observations (Hook et al., 2023).
• Baryogenesis and Astrophysics: Coupling axions to both SM and dark photons alters the thermal and decay history, enabling axion-induced electroweak baryogenesis and opening otherwise excluded fates for the axion (early decay to dark photons suppresses visible channels and relaxes supernova and BBN constraints) (Jeong et al., 16 Oct 2024).

4. Experimental Phenomenology and Probes

Axion–photon–dark photon couplings yield distinctive laboratory signatures:

• Resonant Cavity and Haloscope Searches: Precision experiments such as ADMX and Sikivie haloscopes are sensitive to axion–photon conversion, with their methodologies straightforwardly generalized to cases where axions, photons, and dark photons form a coupled system (Lyapustin, 2011, McAllister et al., 2015). In particular, proper accounting of both electric and magnetic mode couplings becomes essential at high frequency or when using arrays of multiple cavities.
• NMR and Interferometry: Nuclear magnetic resonance setups (e.g., CASPEr-Gradient) monitoring the effective torque on polarized spins can simultaneously probe axion–nucleon, axion–photon, and dark photon couplings, with each signal identifiable by distinct spatial profiles and resonant behaviors. Sensitivities to $$g_{a\gamma\gamma} \sim 2 \times 10^{-16}\ \textrm{GeV}^{-1}$$ and kinetic mixing $\epsilon \sim 3 \times 10^{-16}$ for $m \sim 1\ \mu$eV are attainable if QCD axion nucleon couplings can be probed (Beadle et al., 21 May 2025).
• Collider Signatures: At future $e^+e^-$ colliders (ILC, CEPC, FCC-ee), the coupling $g_{a\gamma'\gamma}$ can be probed via single-photon events with missing energy. Processes such as $e^+e^- \to \gamma' \to a\gamma$ or $e^+e^- \to \gamma^* \to a \gamma'$ (with the undetected axion yielding missing momentum) generate an endpoint in the recoil mass distribution, which gives a direct kinematic method for reconstructing the dark photon mass. Sensitivities down to $g_{a\gamma'\gamma} \sim 10^{-4}\ \textrm{GeV}^{-1}$ are expected for dark photon masses $\sim 10$ GeV (Chen et al., 10 Sep 2025).
• Impact on Existing Constraints and g–2 Anomaly: The presence of the $a\gamma\gamma'$ coupling modifies visible branching ratios of the dark photon (e.g., $\gamma' \to a\gamma$), relaxing limits from visible decays and allowing parameter regions compatible with the muon $(g-2)$ anomaly for dark photon masses above $10$ GeV (Chen et al., 29 May 2024).

5. Theoretical Extensions and Model-Building Implications

The inclusion of axion–photon–dark photon couplings significantly broadens dark sector model-building:

• Non-Abelian Dark Sectors and Global Structure: Gauging a discrete subgroup of the center of non-Abelian dark sector groups and mixing it topologically with the SM modifies the quantization conditions for axion couplings, potentially reducing the lower bound on $g_{a\gamma\gamma}$ by an order of magnitude or more, opening new parameter regions for axion detection (Dierigl et al., 3 Sep 2024).
• Portal Interplay and Relic Abundances: The dark axion portal enables connections between portal couplings (vector, axion, and dark axion) that are not simple products; the coupling strengths can be significant even when traditional portals are suppressed (Kaneta et al., 2016, Kaneta et al., 2017).
• Limits and Novel Decay Channels: Introduction of these mixed couplings requires re-examination of existing laboratory, astrophysical, and cosmological bounds and highlights new pathways for dark matter decay, baryogenesis, and early-universe particle production.

6. Mathematical Structure of the Coupled System

Representative formulas arising in this context include:

Coupling or Process	Expression	Context
Axion–photon–dark photon portal	$$\mathcal{L} \supset -\frac{g_{a\gamma\gamma'}}{2}\, a F_{\mu\nu} \tilde{F}'^{\mu\nu}$$	Fundamental coupling
Dark photon's two-body decay	$$\Gamma(\gamma' \to a\gamma) = \frac{g_{a\gamma\gamma'}^2 m_{\gamma'}^3}{96\pi} (1-m_a^2/m_{\gamma'}^2)^3$$	Collider and cosmological signatures
Freeze-in production collision term	$$\gamma_{fa\to f\gamma'} \simeq g_F(T) \frac{T^6}{\pi^4} \frac{e^2 G_{a\gamma\gamma'}^2}{8\pi}\left(\log\frac{T^2}{m_\gamma^2} + \alpha_{\gamma'}\right)$$	Dark photon relic density
Tachyonic dark photon mode eqn	$$\ddot{A}_{k,\pm} + H \dot{A}_{k,\pm} + [ m_{\gamma'}^2 + \frac{k^2}{a^2} \mp \frac{k}{a}(\beta \dot{\phi}/f_a) ] A_{k,\pm} = 0$$	Early universe production
Collider recoil mass	$M_{\rm recoil}^2 = s - 2\sqrt{s}E_\gamma$	Mass determination for missing-energy events

The signals, constraints, and viable parameter space are highly sensitive to the interplay of these couplings, cosmic history, and the precise nature of the dark sector. The coupling's magnitude, resonance conditions, and degrees of freedom excited (longitudinal vs transverse polarizations) all play critical roles in shaping phenomenology (Hook et al., 2021, Agrawal et al., 2018, Choi et al., 2019).

7. Outlook and Experimental Opportunities

A wide range of present and future experiments—haloscopes, interferometric photon-counting setups, NMR-based nuclear spin resonance, beam dumps, and high-luminosity $e^+e^-$ colliders—directly probe the axion–photon–dark photon parameter space. Theoretical developments on the structure of generalized symmetries, topological mixing, and the dynamics of oscillons and parametric resonance continue to motivate further searches and re-examination of existing data sets.

The search for mixed-sector interactions through axion–photon–dark photon coupling remains an active frontier that connects ultraviolet anomaly structure, dark matter cosmology, and laboratory discovery potential in a tightly interlinked framework.