Axion–Nuclear Spin Couplings

• Axion–nuclear spin couplings are effective interactions defined by derivative and pseudoscalar operators linking axion fields to nucleon spins.
• They manifest as oscillating EDMs, spin-precession effects, and anomalous spin–spin forces, which are probed via NMR and co-magnetometry.
• Enhanced models and refined nuclear structure calculations advance constraints on axion parameters, fostering dark matter and astrophysical research.

Axion–nuclear spin couplings refer to the interaction mechanisms—arising in axion or axion-like particle (ALP) models—by which nuclear spins respond to the presence of axions, typically either as mediators of new spin-dependent forces or as a means for detecting axionic dark matter. These couplings are theoretically motivated by extensions to the Standard Model, notably those intended to solve the strong CP problem, and are characterized at the effective level by derivative or pseudoscalar couplings between the axion field and nucleons. Experimental searches and theoretical treatments span atomic collision experiments, nuclear magnetic resonance (NMR) detection schemes, and precision nuclear, atomic, and astrophysical probes. Key phenomena include anomalous spin–spin interactions, oscillating electric dipole moments (EDMs), time-dependent parity violation, and spin precession induced by the axion wind.

1. Fundamental Interactions and Effective Hamiltonians

The interaction between axions and nuclear spins is described at low energy by dimension-5 operators involving the derivative of the axion field and the nucleon axial-vector current. The generic Lagrangian for nucleons is

$$\mathcal{L}_\mathrm{int} = -\frac{\partial_\mu a}{f_a} \, \bar{\psi} \gamma^\mu \gamma_5 \psi\,,$$

where $a$ is the axion field, $f_a$ is the Peccei–Quinn symmetry-breaking scale, and $\psi$ denotes the nucleon field. This derivative coupling generates effective Hamiltonians and potentials relevant for a variety of processes:

• Spin-dependent monopole–dipole (or dipole–dipole) interactions: These potentials arise from exchange of axions or ALPs. For nucleon pairs,

$$V_\mathrm{m-d}(r) = \frac{\hbar^2 g_s^{(N)} g_p^{(n)}}{8\pi m_n} (\boldsymbol{\sigma} \cdot \hat{\mathbf{r}}) \left( \frac{1}{\lambda r} + \frac{1}{r^2} \right) e^{-r/\lambda}$$

as in (Tullney et al., 2013), where $g_s^{(N)}$ and $g_p^{(n)}$ are scalar and pseudoscalar axion couplings, $m_n$ is the neutron mass, and $\lambda = \hbar/(m_a c)$ is the force range set by the axion mass $m_a$.

• Anomalous spin–spin forces: Experiments have considered new terms in spin-exchange collision potentials between alkali atoms and noble gases, e.g., of Yukawa or dipole–dipole form, as well as velocity- and spin-dependent interactions,

$$V_8(R) \propto \frac{g_A^n g_A^p}{4\pi\hbar c} \frac{\hbar c}{R} (\mathbf{I} \cdot \mathbf{v})(\mathbf{K} \cdot \mathbf{v}) e^{-mcR/\hbar}$$

involving the spins $\mathbf{I}, \mathbf{K}$ and relative velocity $\mathbf{v}$ (Kimball et al., 2010).

• Oscillating EDMs and time-dependent parity violation: The oscillating axion field can mix nuclear states of opposite parity, leading to effects such as

$$d_a(t) \sim -1.8 \times 10^{-19} \left(\frac{a_0}{f_a}\right) \sin(m_a t)\, e\,\mathrm{cm}$$

in $^{199}$Hg, with $a_0$ the axion field amplitude (Stadnik et al., 2013).

• Direct spin–axion coupling and precession: In a covariant description, the spin evolution equation includes a term proportional to the axion gradient:

$$\frac{dS^i}{dT} = \frac{w_{25}}{m} V_j \epsilon^{ikln} S_k p_l$$

where $w_{25}$ is a dimensionless coupling, $V_j = \partial_j \phi$ (the axion gradient), and $p_l, S_k$ are momentum and spin four-vectors (Balakin et al., 2015).

2. Experimental Approaches and Detection Strategies

Experimental searches for axion–nuclear spin couplings exploit the effects of these interactions in both static and time-dependent contexts.

Spin-exchange collision measurements: By comparing measured and calculated cross sections for spin-exchange between Na and $^3$He at the atomic scale ($\sim 10^{-8}$ cm), stringent limits can be set on anomalous spin-dependent nucleon–nucleon forces, providing bounds on coupling constants such as

$g_A^n/\sqrt{4\pi\hbar c} < 2 \times 10^{-3}$

for a hypothetical axial-vector boson of mass $\lesssim 100$ eV (Kimball et al., 2010).

Co-magnetometry and nuclear precession: Ultra-sensitive low-field magnetometers using co-located $^3$He and $^{129}$Xe nuclear spins, detected via SQUIDs, have been used to constrain monopole–dipole interactions mediated by axions or ALPs. Frequency shifts are extracted via

$$\overline{\Delta\nu_\mathrm{sp}} = \frac{\bar{b}_c - \bar{b}_d}{2\pi(1 - (\gamma_{He}/\gamma_{Xe}))}$$

with no observed shift translating into stringent bounds on $|g_s^{(N)} g_p^{(n)}|$ as a function of range $\lambda$ (Tullney et al., 2013).

Precision NMR/dark matter searches: CASPEr and similar experiments employ NMR techniques to identify oscillating torques on nuclear spins arising from axion dark matter. The predicted signal is a resonantly enhanced transverse magnetization when the Larmor frequency matches the axion Compton frequency,

$\omega_0 = \gamma B_0 = m_a/\hbar$

with detection bandwidths tuned for $m_a$ scanning (Garcon et al., 2017, Wang et al., 2017).

Solid-state and quantum sensors: Methods such as nitrogen-vacancy (NV) center nuclear spin magnetometry are being implemented for axion-nuclear coupling searches, exploiting long nuclear spin coherence times to probe low axion masses ($m_a \lesssim 4\times 10^{-13}$ eV) (Chigusa et al., 9 Jul 2024).

Superfluid $^3$He and bosonic magnon modes: Experiments using magnon Bose–Einstein condensates in $^3$He (A₁ phase) or the homogeneous precession domain (HPD) leverage the axion wind effect, where axions induce shifts in the precession frequency of a large-amplitude, coherent NMR signal (Gao et al., 2022, Chigusa et al., 2023).

3. Theoretical Modeling and Interpretation of Constraints

Translation from experimental observables to bounds on axion couplings requires careful modeling of both the nuclear spin content and operator structure:

• Nuclear spin content: Determining the fractions $\sigma_p, \sigma_n$ of spin carried by protons and neutrons in a given nucleus is essential. Semi-empirical models are insufficient for precise bounds; large-scale shell-model calculations provide improved estimates (Kimball, 2014). The effective coupling in an atom can be recast as

$\chi_N = \chi_p \sigma_p + \chi_n \sigma_n$

and the mapping from measured precession frequency shifts to the underlying product of coupling constants $g_p g_s$ must incorporate these content factors.

• Monopole–dipole and dipole–dipole potentials: For a spin-dependent force mediated by axion exchange, the monopole–dipole interaction has the form

$$V_{m\text{--}d}(r) = \frac{g_p^X g_s^Y \hbar}{8\pi m_X c} (\mathbf{S}_X \cdot \hat{\mathbf{r}})\left(\frac{1}{r\lambda}+\frac{1}{r^2}\right) e^{-r/\lambda}$$

and the dipole–dipole term is

$$V_{d\text{--}d}(r) = \frac{g_p^X g_p^Y \hbar^2}{16\pi m_X m_Y c^2 r^3} [\mathbf{S}_X \cdot \mathbf{S}_Y - 3(\mathbf{S}_X \cdot \hat{\mathbf{r}})(\mathbf{S}_Y \cdot \hat{\mathbf{r}})]$$

(Kimball, 2014). Updated nuclear modeling revises the mapping between observed signals and exclusion limits on the parameters $g_p, g_s$.

• Resonant enhancement: Axion-induced effects in atoms and nuclei—such as oscillating EDMs—are resonantly enhanced when the energy splitting $\Delta$ between parity-opposite states matches $m_a$, leading to mixing amplitudes scaling as $1/(\Delta^2 - m_a^2)$ (Stadnik et al., 2013).

4. Velocity and Gravitational Dependence; New Coupling Structures

Beyond static couplings, several papers explore additional signatures:

• Velocity-dependent terms: Effective interactions involving spin and velocity, e.g. $$(\mathbf{I}\cdot\mathbf{v})(\mathbf{K}\cdot\mathbf{v})$$, are constrained for the first time at the atomic scale for long-range bosons with $m\lesssim 1$ eV (Kimball et al., 2010).
• Gravitational field effects: The Earth's gravitational field distorts the axion field, producing terms in the effective Hamiltonian proportional to $\mathbf{g} \cdot \boldsymbol{\sigma}$, which are distinct from the usual "axion wind" term ($\mathbf{p}_a\cdot\boldsymbol{\sigma}$) (Stadnik et al., 2013).
• Torsion gravity links: Constraints on nuclear spin–dependent forces can be mapped onto parameters describing torsion gravity in Riemann–Cartan spacetimes, with bounds on the dimensionless constant $\beta$ as a function of interaction range (Kimball et al., 2010).

5. Novel Model-Enhancement Mechanisms and Theoretical Implications

Model-building efforts can selectively enhance axion-nucleon couplings:

• Clockwork/DFSZ-like models: By distributing Peccei–Quinn charges in an array of Higgs doublets and applying clockwork mechanisms, the coupling to specific quark generations (and thus to nucleons) can be exponentially enhanced without increasing $m_a$. This leads to significantly higher axion–nuclear spin interaction strengths, potentially bringing them into the reach of current and next-generation experiments (Darmé et al., 2020).
• Astrophysical implications: Enhanced couplings influence neutron star environments, potentially explaining anomalous hard X-ray emission via axion emission and subsequent conversion in magnetic fields.

This suggests that revised or engineerably enhanced axion–nuclear couplings expand the phenomenological landscape, motivating both experimental and astrophysical searches in a broader parameter space.

6. Distinguishing Signal Channels and Complementarity

Current and prospective experiments are sensitive to multiple types of dark sector interactions:

• Equivalence of effective and real fields: Nuclear magnetic resonance–based axion searches (e.g., CASPEr–Gradient) respond identically to an effective axion-induced magnetic field ($\mathbf{B} \propto \nabla a$) or a real magnetic field produced by kinetically mixed dark photons or axion–photon coupling in a background field—distinctions arise from spatial mode structure and scan strategies (Beadle et al., 21 May 2025).
• Signal discrimination: The spatial profile (homogeneous for axion–nucleon coupling; inhomogeneous for dark photons or axion–photon conversion in cavities) permits systematic separation of hypotheses based on sample placement and field symmetry.
• Multi-channel sensitivity: Achieving sensitivity to the QCD axion parameter space for $g_{aN}$ ensures simultaneous reach to kinetic mixings as low as $\epsilon \sim 3 \times 10^{-16}$ (dark photon) and $g_{a\gamma\gamma} \sim 2 \times 10^{-16}$ GeV$^{-1}$ (axion–photon) for $m_a\sim1\,\mu$eV, highlighting the multipronged impact of these experimental strategies.

7. Summary Table: Operator Structures and Key Experimental Constraints

Interaction Type	Operator/Potential (Schematic)	Leading Experimental Constraints
Derivative coupling	$(\partial_\mu a/f_a)\bar\psi\gamma^\mu\gamma^5\psi$	$|g_A^n/\sqrt{4\pi\hbar c}| < 2\times 10^{-3}$ (Kimball et al., 2010)
Monopole–Dipole	$$V_{m\text{--}d}\propto (\mathbf{S}\cdot\hat{\mathbf{r}})e^{-r/\lambda}/r$$	SQUID co-magnetometry (Tullney et al., 2013, Kimball, 2014)
Dipole–Dipole	$$V_{d\text{--}d}\propto [\mathbf{S}_X\cdot\mathbf{S}_Y - 3(\mathbf{S}_X\cdot\hat{\mathbf{r}})(\mathbf{S}_Y\cdot\hat{\mathbf{r}})]/r^3$$	NMR/atomic clock ensembles (Kimball, 2014, Wang et al., 2022)
Oscillating EDM	$d_a(t)\sim-[\text{prefactor}] (a_0/f_a) \sin(m_at)$	$^{199}$Hg, $^{225}$Ra, neutron EDM (Stadnik et al., 2013, Abel et al., 2017)
Axion wind (velocity)	$$H_\mathrm{int}(t)\sim (C_N a_0/2f_a)\sin(m_a t) \boldsymbol{\sigma}_N\cdot\mathbf{p}_a$$	UCN, NMR (Abel et al., 2017, Gao et al., 2022)
Torsion gravity	$V_{T}\propto\beta$ (torsion parameter)	Na–$^3$He collision bounds (Kimball et al., 2010)

Note: Table columns are kept concise to conform with format guidelines. Detailed formulas and reference numbers are included above.

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The comprehensive paper of axion-nuclear spin couplings integrates theoretical frameworks for spin-dependent interactions, stringent experimental constraints across atomic and nuclear systems, the necessity of nuclear structure modeling, and the broadening of models to include enhanced couplings or multi-channel dark sector detection. These couplings are central to both the direct laboratory search for axionic dark matter and the indirect constraints from precision atomic, molecular, and astrophysical observations. The variety of operator structures, experimental schemes, and methods for discriminating among possible signals ensures that this research area remains at the intersection of particle, atomic, and astrophysics.