Dark Matter–Electron Scattering Cross-Sections

• Dark matter–electron scattering cross-sections quantify the probability of dark matter interacting with electrons, key for sub-GeV candidate detection.
• These cross-sections depend on microscopic parameters, including velocity, momentum transfer, mediator type, and environmental effects.
• Accurate modeling involving many-body effects and excitonic enhancements is essential for interpreting data from direct, astrophysical, and cosmological experiments.

Dark matter–electron scattering cross-sections quantify the probability per unit flux for dark matter (DM) to interact elastically or inelastically with electrons via various interaction channels. These cross-sections are central to both the phenomenology of direct-detection experiments and to the astrophysical and cosmological consequences of dark matter, particularly for light DM candidates (sub-GeV) where nuclear recoils are kinematically suppressed. A rigorous treatment requires careful attention to the microscopic theory of DM–electron interactions, their environment-dependent structure (in atomic, condensed matter, or astrophysical contexts), and their impact on the distribution and evolution of matter in the Universe.

1. Formalism: Microscopic Cross-Section Definitions

The dark matter–electron scattering cross-section is conventionally described via the differential cross section $d\sigma/d\Omega$ or $d\sigma/dE_e$ (with respect to either angle or recoil energy), and occasionally through the momentum-transfer cross-section: $$\sigma_{\rm MT} \equiv \int d\Omega\,\frac{d\sigma}{d\Omega}\,(1-\cos\theta),$$ which weights by the longitudinal momentum transfer [(Nguyen et al., 2021)].

For nonrelativistic or low-energy elastic scattering, the cross section can be expressed as: $\sigma_{\rm MT}(v) = \sigma_0 v^n,$ where $\sigma_0$ is a reference cross-section and $n$ parametrizes the velocity dependence, capturing the physics for contact, dipole, or light-mediator models ($n \in \{-4,-2,0,2,4,6\}$) [(Nguyen et al., 2021)]. More generally, the differential rate of events in a material is given by

$$\frac{dR}{dE_e} \propto \int d^3v_\chi\,f_\chi(v)\,v\,\frac{d\sigma}{dE_e}(v, E_e).$$

For composite DM (e.g., dark atoms), the internal structure—encoded in partial-wave phase shifts, binding energies, and resonance features—enters explicitly in determining $d\sigma/dE_e$ [(Cline et al., 2013)].

2. Physical Dependencies and Parameter Scaling

The cross-section depends sensitively on both microphysical parameters and environmental factors:

• Interaction Mediation: The Lorentz structure, type of mediator (scalar, pseudoscalar, vector, axial), and the nature of the underlying operator (e.g., $\bar{\chi}\gamma^\mu\chi\,\bar{e}\gamma_\mu e$) set the fundamental scaling, form factors, and potential momentum or velocity dependencies [(Wu et al., 2022, Dhyani et al., 6 Mar 2025)].
• Mediators and Form Factors: For models with light mediators, the cross-section often contains a form factor $|F_{\rm DM}(q)|^2$ with $q$ the momentum transfer, e.g., $|F_{\rm DM}(q)|^2 \sim (\alpha m_e/q)^n$, leading to long-range enhancements at low $q$ [(Radick et al., 2020, Nguyen et al., 2021)].
• Reduced Mass and Energy Scaling: The reduced mass $\mu_{\chi e}$ and the ratio $m_\chi / m_e$ control both the kinematic threshold and the per-event deposited energy. For $m_\chi \ll m_e$, the cross-section sensitivity scales as $m_\chi^2$ as a consequence of the phase-space and recoil spectrum [(Shang et al., 13 Mar 2024)].
• Composite DM Structure: For atomic dark matter, the scattering cross-section can be "designer"—tuned by varying internal mass ratios (e.g., $R = m_{p'}/m_{e'}$), leading to resonance structure, poles, and zeros in the scattering length $a_s$, with nontrivial consequences for low-energy cross sections and response [(Cline et al., 2013)].

3. Environmental and Astrophysical Contexts

3.1 Galactic and Substructure Velocity Distributions

The event rate is highly sensitive to the assumed DM velocity distribution $f_\chi(v)$ and the target mean inverse speed $\eta(v_{\rm min})$: $$\eta(v_{\rm min}) = \int_{v \geq v_{\rm min}} \frac{f_\chi(v)}{v} d^3v,$$ where $v_{\rm min}$ is set by the energy threshold for observing a recoil [(Radick et al., 2020, Maity et al., 2022)].

• Variations in the circular speed, escape speed, distribution shape (Maxwell-Boltzmann, Tsallis, or empirical Milky Way models), and the possible presence of dark matter substructures (streams, debris flows) can alter the predicted rate by factors ranging from $\mathcal{O}(1)$ to orders of magnitude, with the high-velocity tail especially impactful for electron recoils due to the need to exceed binding energies or detection thresholds [(Maity et al., 2022)].

3.2 Terrestrial and Astrophysical Attenuation

• Terrestrial Stopping: Underground direct detection of DM–electron scattering is limited by "nuclear stopping," where DM particles must traverse Earth's crust and lose energy via multiple scatterings with nuclei, imposing an upper bound on the cross-section reach and opening only a finite window for detectable masses and couplings in subsurface laboratories [(Emken et al., 2017)].
• Solar Capture and Neutrino Flux Constraints: In astrophysical settings (e.g., the Sun), elastic scattering with electrons enables gravitational capture of DM, with subsequent annihilations producing detectable neutrino signatures; the capture rate depends on both the momentum dependence of the cross section and the solar temperature/electron density profile [(Garani et al., 2017, Maity et al., 2023, Krishna et al., 10 Mar 2025)].
• Molecular Cloud Ionization: Strongly interacting DM can induce ionization in molecular clouds, offering complementary constraints in regimes where terrestrial detectors lose sensitivity due to overburden or high backgrounds, provided the energy transferred by DM can overcome binding energies (e.g., $\sim15\,\mathrm{eV}$ in H$_2$) [(Prabhu et al., 2022)].

4. Material and Many-Body Effects

A precise theoretical prediction for DM–electron scattering in condensed matter targets (semiconductors, insulators, scintillators) demands an accurate computation of the dielectric response:

• Energy-Loss Function (ELF): The scattering rate per unit mass involves the imaginary part of the inverse dielectric function ($-\mathrm{Im}[1/\epsilon(\omega, \vec{k})]$).

$$R \propto \int d^3v\,f_\chi(v) \int d^3k\,|F_\phi(k)|^2 \int d\omega\,(-\mathrm{Im}[1/\epsilon(\omega, \vec{k})])\delta(\omega + k^2/2m_\chi - \vec{k}\cdot\vec{v})$$

[(Taufertshöfer et al., 18 Jul 2025)].

• Excitonic Effects: In some materials (e.g., NaI), electron-hole interactions (excitons) can dramatically enhance the ELF near the band gap, raising the effective DM–electron scattering rate by nearly an order of magnitude compared to predictions neglecting such effects (as in random-phase approximation). For others (e.g., GaAs), with small exciton binding, excitonic enhancement is negligible [(Taufertshöfer et al., 18 Jul 2025)].
• Atomic and Molecular Shell Structure: In direct detection experiments, target atom shell structure and electronic binding energies govern the accessible recoil energies and, via the ionization form factor, modulate the cross-section sensitivity in the keV and sub-keV regime [(Shang et al., 13 Mar 2024)].

5. Sensitivity in Experimental and Observational Settings

Experimental and cosmological constraints on DM–electron scattering cross-sections span many orders of magnitude in cross section and DM mass, drawing on a variety of methods:

• Direct Detection: Underground experiments (XENON10/100/1T, DAMIC, SENSEI, PandaX-4T, etc.) probe cross-sections down to $\sim10^{-40}$–$10^{-44} \,\mathrm{cm}^2$ for DM masses from about 10 eV to several GeV, with sub-MeV reach enabled by cosmic-ray boosting or ultra-low threshold technologies [(Shang et al., 13 Mar 2024, Emken et al., 2017)].
• Astrophysical and Neutrino Telescopes: Neutrino fluxes from the Sun (IceCube, DeepCore, Super-Kamiokande, Hyper-Kamiokande) set strict limits on DM–electron scattering via solar capture and annihilation-induced neutrinos, outperforming direct detection in the mass range 4–200 GeV by reaching cross sections down to $10^{-40}$–$10^{-39}\,\mathrm{cm}^2$ [(Maity et al., 2023, Krishna et al., 10 Mar 2025)].
• Cosmological Constraints: Analysis of the CMB (Planck 2018) and the satellite abundance (DES, Pan-STARRS1) places powerful, complementary bounds on the momentum-transfer cross section (e.g., $\sigma_{\rm MT} = \sigma_0 v^n$) by requiring that structure formation is not suppressed beyond observed levels. These constraints are uniquely sensitive at sub-MeV masses and fill the region of high cross section inaccessible to direct detection [(Nguyen et al., 2021, Dhyani et al., 6 Mar 2025)].
• Energy-Dependent Scattering: In both blazar-boosted and cosmic-ray–boosted scenarios, the intrinsic energy dependence of the scattering amplitude (as inherited from the S-matrix and the Lorentz structure of the interaction vertex) can enhance or suppress predicted event rates by orders of magnitude compared to constant cross-section approximations [(Bardhan et al., 2022, Bhowmick et al., 2022)].

Experiment/Context	Mass Sensitivity	Cross-Section Sensitivity	Dominant Limitation
Direct detection (Xe/Si)	10 eV – GeV	$\sim$10$^{-40}$–$10^{-44}$ cm²	Threshold, overburden, terrestrial stopping
Neutrino telescopes	4 GeV – 10$^5$ GeV	$\sim$10$^{-40}$–$10^{-39}$ cm²	Equilibrium/annihilation channel
Cosmology (CMB, satellites)	10 keV – TeV	Model and velocity law dependent	Energy injection, small-scale power
Astrophysical (MCs, Sun)	>4 MeV or >10 GeV	Complementary to others	Target composition, energy thresholds

6. Theoretical and Model-Building Implications

• Spin-Dependent versus Spin-Independent: For scalar or vector DM mediated by pseudoscalar or axial-vector couplings, "spin-dependent" scattering is characterized by $p$-wave (momentum-dependent) matrix elements, potentially enhancing the cross section by orders of magnitude relative to the $s$-wave (momentum-independent) SI contributions at small $q$ [(Wu et al., 2022)].
• Composite Dark Matter and Resonance Structure: Internal structure in atomic or molecular DM can result in highly nontrivial energy- and mass-dependent cross sections, including resonances (poles/zeros in atomic scattering lengths) and novel dissipation channels (e.g., rotational excitations in dark molecules exceeding certain mass thresholds, leading to dissipative self-interactions) [(Cline et al., 2013)].
• Operator-Level Correlations: In ultraviolet (model) completions involving heavy mediators, effective operators for DM–electron coupling generically predict both annihilation and scattering, so CMB bounds on annihilation and drag must be simultaneously considered; including both effects can alter limits by 10–15% compared to considerations of only one channel [(Dhyani et al., 6 Mar 2025)].

7. Outstanding Issues, Future Prospects, and Experimental Directions

• Velocity- and Momentum-Dependent Rates: Experimental projections must incorporate realistic (non-Maxwellian) velocity distributions and account for substructure to accurately assess sensitivity, especially for low-mass DM and for interpreting statistical anomalies [(Maity et al., 2022)].
• Excitonic and Strongly-Correlated Electron Effects: For bulk materials, accurate predictions demand many-body methods (Bethe–Salpeter equation) rather than single-particle approximations (RPA), particularly for wide-band-gap or strong-exciton-binding scintillators [(Taufertshöfer et al., 18 Jul 2025)].
• Astrophysical Diagnostics: Measurements of ionization rates in molecular clouds and next-generation space- and balloon-based detectors will further limit, or potentially reveal, strongly interacting DM components in regimes inaccessible to underground laboratories [(Prabhu et al., 2022)].
• Complementarity of Methods: Exploiting the synergy between cosmological surveys, neutrino telescopes, direct searches, and precision astrophysical probes is essential to map the full parameter space of DM–electron interactions, especially as theoretical models include velocity-dependent operators, composite structures, and new sources of dissipative channels.

In total, dark matter–electron scattering cross-sections are now constrained over a vast mass and cross-section range through a combination of direct detection, cosmological, and astrophysical observations. Theoretical modeling must faithfully account for the full amplitude and environmental structure to make precise predictions that match the rapidly improving experimental sensitivity and the increasing richness of observational data.