Spin-Dependent Scattering of Sub-GeV Dark Matter

• The paper explains that spin-dependent scattering involves dark matter coupling to nucleon and electron spins via axial-vector and pseudoscalar portals.
• It details how differential cross sections, kinematic thresholds, and many-body effects in nuclear, atomic, and material contexts shape detection strategies.
• The study highlights experimental techniques and constraints from astrophysical, collider, and direct detection searches that refine the sub-GeV dark matter parameter space.

Spin-dependent scattering of sub-GeV dark matter (DM) encompasses the class of interactions in which DM couples to Standard Model (SM) spin degrees of freedom—typically electron or nucleon spins—mediated by various operators arising from pseudoscalar or axial-vector portals. These processes are central to formulating, modeling, and interpreting direct-detection searches for light DM candidates, where standard spin-independent (SI) approaches often lose efficiency due to kinematic thresholds. The field synthesizes nonrelativistic effective theory, nuclear and atomic many-body response calculations, condensed matter physics, and stringent astrophysical and accelerator constraints on mediator couplings.

1. Theoretical Foundations of Spin-Dependent Scattering

Operator Structures and Mediator Models

Spin-dependent (SD) interactions generically arise through axial-vector or pseudoscalar couplings, in contrast to the scalar/vector mediators contributing to SI scattering. At the Lagrangian level, three principal portals are considered for DM–nucleon SD scattering, particularly relevant below the GeV scale (Gori et al., 12 Jun 2025, Ramani et al., 2019):

• Scalar ϕ with pseudoscalar nucleon couplings:

$$\mathcal{L}_\phi = \phi \left[g_\chi\,\bar{\chi}\chi + g_p\,\bar{p}\gamma^5 p + g_n\,\bar{n}\gamma^5 n\right]$$

• Axion-like pseudoscalar $a$:

$$\mathcal{L}_a = a \left[g_\chi\,\bar{\chi}\gamma^5\chi + g_p\,\bar{p}\gamma^5 p + g_n\,\bar{n}\gamma^5 n\right]$$

• Axial-vector $A'_\mu$:

$$\mathcal{L}_{A'} = A'_\mu \left[g_\chi\,\bar{\chi}\gamma^\mu\gamma^5\chi + g_p\,\bar{p}\gamma^\mu\gamma^5 p + g_n\,\bar{n}\gamma^\mu\gamma^5 n\right]$$

Analogous operator structures describe SD DM–electron and DM–molecule interactions, including non-relativistic effective field theory (NR-EFT) bases that generalize the treatment to a full set of Galilean-invariant operators (Giffin et al., 13 Nov 2025, Liu et al., 2021).

Differential Cross Section and Kinematics

The canonical SD WIMP–nucleus cross section at zero momentum transfer is (Abdelhameed et al., 2019):

$$\sigma_N^{\rm SD} = \frac{4}{\pi} \mu_N^2\frac{\left(a_p \langle S_p\rangle + a_n \langle S_n\rangle\right)^2}{2J+1}$$

with $\mu_N$ the reduced mass, $J$ nuclear spin, $\langle S_{p,n}\rangle$ expectation values, and $a_{p,n}$ effective couplings.

Mediator type and mass induce characteristic momentum dependencies in the cross sections; for instance, axial-vector exchange yields:

$$\frac{d\sigma}{dE_R} \propto \frac{1}{v^2}\left(\frac{q^2}{(q^2 + m_{\rm med}^2)^2}\right)\quad\text{(pseudoscalar)}$$

or a contact term for heavy mediators (Gori et al., 12 Jun 2025, Ramani et al., 2019). The form factor structure is further modified by many-body effects, especially in the molecular and condensed matter context (Essig et al., 2019, Trickle et al., 2019).

2. Nuclear, Atomic, and Material Response

Nucleus-Level: Shell Model and Chiral EFT

The nuclear response in SD processes is encoded in structure factors $S_{N}(q)$, dependent on the nuclear spin content and form factors derived from shell-model or chiral EFT calculations (Wang et al., 2021). For a nuclear ground state with spin $J$,

$$S_N(0) = \frac{2J+1}{4\pi J}(J+1) |(a_0+a_1)\langle S_p\rangle + (a_0-a_1)\langle S_n\rangle|^2$$

Higher-order corrections include momentum-dependent terms and two-body currents relevant at larger $q$ (Wang et al., 2021).

Electron-Level: Atomic Many-Body Calculations

For DM–electron SD scattering, the atomic ionization response is central (Liu et al., 2021, Wu et al., 2022). The leading NR operator is $\vec{S}_\chi \cdot \vec{S}_e$, with many-body effects encoded in the atomic transition matrix elements:

$$R_{\rm SD}^{\rm ion}(T,q) = \sum_{I,F}\sum_{j=1}^3 |\langle F|\sum_{i=1}^Z e^{i\mathbf{q}\cdot\mathbf{r}_i}\sigma^D_{i,j}|I\rangle|^2\delta(E_F-E_I-T)$$

Relativistic corrections—especially spin–orbit coupling (SOC)—modify the recoil-energy spectrum and scale the response away from a constant SD:SI ratio (Liu et al., 2021, Chen et al., 2022).

Solid-State and Molecular Targets

Collective excitations such as magnons in magnetic materials (Trickle et al., 2019) or molecular vibrational/rotational modes (Essig et al., 2019) enable SD sensitivity far below traditional nuclear-recoil thresholds. For example, the magnon structure factor in a ferrimagnet (e.g., Y$_3$Fe$_5$O$_{12}$) governs the transition rate for DM-induced single-magnon creation, while molecular form factors set the vibrational excitation probabilities in molecular gases.

SOC in materials supports strong SD–electron coupling, modifying coherent scattering rates and sensitivity boundaries for both electrons and phonons (Chen et al., 2022, Giffin et al., 13 Nov 2025).

3. Experimental Techniques and Sensitivities

Direct Detection Approaches

• Cryogenic targets: Experiments deploying Li$_2$MoO$_4$ or similar targets use the substantial nuclear spin of $^7$Li or $^{19}$F to enhance SD sensitivity in the GeV regime (Abdelhameed et al., 2019).
• Noble liquid detectors: Liquid xenon experiments (XENON10/100/1T, LZ) provide leading bounds for DM–nucleus SD interactions, with sensitivity depending on the unpaired nucleon (e.g., unpaired neutron in Xe, leading to stronger neutron-coupling limits) (Wang et al., 2021, Li et al., 2022).
• Ultracold molecules and gases: Ro-vibrational molecular excitation in optically or cryogenically cooled CO, HF, HBr, or ScH extends SD reach into the sub-GeV mass window (Essig et al., 2019).
• Solid-state detectors: Material platforms with strong SOC, magnons in ferrimagnets, or phononic excitation in crystals provide access to sub-MeV masses by leveraging macroscopic coherence and collective responses (Chen et al., 2022, Trickle et al., 2019).

Signal Modeling and Operator Mapping

Practically, the differential event rate is schematically:

$$\frac{dR}{dE} = \frac{\rho_\chi}{m_\chi}\,\sigma \times \text{[Material Response]} \times \eta(v_\text{min})$$

with $\eta(v_\text{min})$ encoding the halo velocity integral. For inelastic scenarios, exothermic and endothermic transitions affect threshold kinematics, with exothermic scattering yielding lowered $v_\text{min}$ and stronger exclusion (Li et al., 2022).

Spin-dependent signals can be further disentangled via spectral-shape analysis and, in some materials, through directional or polarization-sensitive observables (i.e., magnon emission) (Giffin et al., 13 Nov 2025, Trickle et al., 2019).

4. Constraints: Astrophysics, Colliders, and Direct Detection

Terrestrial and Astrophysical Limits

The allowed SD cross section for sub-GeV DM is tightly restricted by complementary probes:

• Stellar cooling and supernova bounds: Light mediators are strongly constrained by supernova energy-loss limits, with $g_N$ coupling to nucleons bounded to $10^{-9}$–$10^{-4}$ depending on $m_{\text{med}}$ (Ramani et al., 2019, Gori et al., 12 Jun 2025).
• Meson decay and beam dump experiments: Rare decays such as $K\to\pi\phi$ and beam dump searches (e.g., CHARM, E137) exclude large regions of mediator parameter space.
• Self-interaction constraints (SIDM): Dark matter self-scattering is limited to $\sigma_V/m_\chi \lesssim 1\ \text{cm}^2/\text{g}$ at galaxy cluster velocities, setting upper limits on $g_\chi$ (Gori et al., 12 Jun 2025, Ramani et al., 2019).
• Direct detection: Current direct-detection experiments (PICO, XENON1T, SuperCDMS) probe $\sigma_{\rm SD} \lesssim 10^{-41}$–$10^{-44}$ cm$^2$ for $m_\chi \gtrsim 100$ MeV, with molecular and magnonic platforms projected to reach below $10^{-38}$ cm$^2$ at $m_\chi\sim10$–$100$ MeV (Essig et al., 2019, Trickle et al., 2019).

A narrow window in parameter space, especially for a scalar mediator in the "trapping window" (300 keV $\lesssim m_\phi \lesssim$ 100 MeV), survives these bounds and may be within future experimental reach (Gori et al., 12 Jun 2025, Ramani et al., 2019).

5. Material and Kinematic Effects

Mediator and Target Dependence

The scattering rate can exhibit thresholds and enhancements based on kinematic matching between mediator mass, DM mass, and the energy scale of available excitations (e.g., meV phonons, vibrational transitions, magnon modes) (Chen et al., 2022, Trickle et al., 2019). Materials with strong spin–orbit coupling or magnonic order can exhibit amplified or distinctive SD responses, and theoretical modeling must incorporate the full electronic or lattice Hamiltonian, including SOC and anisotropy (Giffin et al., 13 Nov 2025, Chen et al., 2022).

Inelastic and Boosted Scattering

In inelastic DM scenarios (mass splitting $\delta$), exothermic ($\delta<0$) and endothermic ($\delta>0$) events have sharply differing rate profiles and sensitivity to the high-velocity tail of the DM halo model (Li et al., 2022). "Boosted" DM—DM particles with kinetic energies well above virial expectation (e.g., cosmic-ray upscattering)—requires a relativistic extension including time component nuclear currents, which can enhance sensitivity at high $q$ (Wang et al., 2021).

Comparative Table: Experimental Projections

Technique	Mass Range Probed	SD Sensitivity Achievable
Noble liquid (Xe)	$\sim$1 MeV – 10 GeV	$\sigma_n^{\rm SD} \sim 10^{-33}$–$10^{-44}$ cm$^2$ (Wang et al., 2021, Liu et al., 2021)
Molecular gas	0.2 MeV – 1 GeV	$\sigma_n^{\rm SD} \sim 10^{-38}$ cm$^2$; $\sigma_p^{\rm SD} \sim 10^{-37}$ cm$^2$ (Essig et al., 2019)
Magnon detectors	10 keV – 10 MeV	$\overline\sigma_e \sim 10^{-30}$–$10^{-31}$ cm$^2$ (Trickle et al., 2019)

6. Outlook and Open Problems

• Parameter space: Only narrow windows in SD cross section–mass space remain accessible to future direct detection given mediator, self-interaction, and astrophysical bounds, particularly for scalar mediators in the tens–hundreds of MeV regime (Gori et al., 12 Jun 2025, Ramani et al., 2019).
• Material innovation: Further theoretical and experimental developments in materials with strong SOC, low-threshold magnetic order, or exotic quasi-particles are critical for opening new detection avenues in the sub-MeV regime.
• Operator discrimination: Ongoing research aims to disentangle SD from SI contributions via spectral, polarization, and/or directional analysis, including systematics due to relativistic and many-body corrections (Liu et al., 2021, Giffin et al., 13 Nov 2025).
• Dark sector model space: Extensions involving higher-spin DM, non-standard mediators, and inelastic or composite DM are active loci of investigation, with direct-detection bounds sensitive but often insufficient to distinguish spin or mediator origin (Wu et al., 2022).

Spin-dependent scattering of sub-GeV dark matter is thus a nexus for precision nuclear, atomic, condensed matter, and astrophysical calculations, and advances in both theory and low-threshold detector technology are essential for closing the remaining windows in SD parameter space.