Dark Matter–Nucleon Scattering Cross-Section

• Dark-matter–nucleon scattering is a measure of the probability for dark matter particles to interact with nucleons under various theoretical and experimental conditions.
• Calculations employ effective field theories, operator matching, and nucleon matrix elements to characterize both spin-independent and spin-dependent interactions.
• Experimental strategies range from direct detection to astrophysical observations, enabling constraints over vast cross-section and mass scales.

Dark-matter–nucleon scattering cross-section (σ_χN) quantifies the effective probability for a dark matter (DM) particle to elastically or inelastically scatter from a nucleon. This parameter underlies both theoretical model-exclusion and experimental search strategies across the full mass range of viable DM candidates. The cross-section is not a single fixed number but a function of DM mass, underlying mediator structure, velocity and momentum transfer, and the nuclear environment. Its determination underpins efforts to probe, constrain, or discover non-gravitational DM interactions with ordinary matter.

1. Theoretical Framework for DM–Nucleon Scattering

The standard formalism for DM–nucleon scattering is built from effective field theory, integrating out heavy mediators to yield local operators coupling DM bilinears to quark and gluon currents. The generic non-relativistic effective Lagrangian is

$$\mathcal{L}_{\rm eff} = \sum_{q} C_S^q\,m_q\,\overline{\chi}\chi\,\overline{q}q + C_{AV}^q\,\overline{\chi}\gamma^\mu\gamma_5\chi\,\overline{q}\gamma_\mu\gamma_5 q + \sum_{i=1,2} C_{T_i}^q\,\mathcal{O}^q_{T_i} + C_S^g\,\frac{\alpha_s}{\pi}\,\overline{\chi}\chi\,G^A_{\mu\nu}G^{A\mu\nu} + ...$$

where $\chi$ denotes the DM field, $q$ is the quark flavor, $G^A_{\mu\nu}$ is the gluon field strength, and $C_i$ are Wilson coefficients encoding short-distance physics (Hisano et al., 2015).

From this, two classes of cross-section are central:

• Spin-independent (SI), dominated by scalar and twist-2 couplings to quarks and gluons.
• Spin-dependent (SD), governed by axial-vector couplings to nucleon spin.

For SI scattering, the per-nucleon zero-momentum-transfer cross-section is

$$\sigma^{N}_{\rm SI} = \frac{4}{\pi}\,\mu_N^2\,|f_N|^2$$

with reduced mass $\mu_N = m_\chi m_N/(m_\chi + m_N)$ and $f_N$ collecting all hadronic matrix elements and Wilson coefficients (Hisano et al., 2015, Hisano et al., 2011, Hill et al., 2011).

The SD cross-section for Majorana or vector DM is

$$\sigma^{N}_{\rm SD} = \frac{12}{\pi}\,\mu_N^2\,|a_N|^2$$

with $a_N$ reflecting the nucleon's spin content $a_N = \sum_q C_{AV}^q \Delta q_N$ where $\Delta q_N$ are nucleon spin fractions.

Alternative mediators, such as pseudoscalar or axial-vector, can introduce strong velocity and/or momentum suppression:

• For pseudoscalar mediation, $\sigma_{\chi N} \propto q^2$, yielding strong suppression at low recoil (Azevedo et al., 2018, Maity et al., 2021).
• For axial-vector mediation, $\sigma_{\chi N} \propto v^2$, again reducing the rate in terrestrial detectors (Maity et al., 2021).

2. Cross-Section Calculation: Methodologies and Model Implementations

The computation of $\sigma_{\chi N}$ requires several sequential steps:

Operator Matching and Wilson Coefficients

Matching at the UV scale (typically the heavy mediator or new-physics scale) determines $C_i$ via tree-level or loop-level diagrams. For example:

• For a Majorana DM coupled via colored scalar mediators, $C_{S,AV,T_i}$ are functions of mediator couplings and mass splittings (Hisano et al., 2015).
• For EW multiplet DM (WIMP), 1-loop $Z/h$-exchange induces $C_i$ with partial cancellation among diagrams, yielding SI cross-sections as low as $10^{-48}$–$10^{-46}$ cm² for TeV-scale masses (Hisano et al., 2011, Hill et al., 2011).

Renormalization Group Evolution

RGEs are used to run Wilson coefficients from the mediator scale down to the hadronic scale where nucleon matrix elements are evaluated. The twist-2 (moment) operator coefficients, in particular, experience significant QCD mixing and evolution (Hisano et al., 2015, Hill et al., 2011).

Nucleon Matrix Elements

Scalar couplings employ nucleon $\sigma$-terms $f_{T_q}$, gluon contributions via the QCD trace anomaly, and twist-2 moments using parton distribution integrals at specified scales (Hisano et al., 2015, Nagata, 2012). Recent lattice QCD provides $f_{T_u}^{(p)}\simeq0.019$, $f_{T_d}^{(p)}\simeq0.027$, $f_{T_s}^{(p)}\simeq0.009$, and $f_{TG}^{(p)}\simeq0.945$ (Hisano et al., 2015).

Nuclear Enhancement and Scaling

For heavy nuclei, Born-level coherence yields

$$\sigma_{\chi A} = A^2\,(\mu_A^2/\mu_N^2)\,\sigma_{\chi N}$$

saturating toward the $A^4$ scaling in the large DM-mass limit ($\mu_A\approx A m_N$) (Digman et al., 2019, Bramante et al., 2018). However, this scaling fails for $\sigma_{\chi N}\gtrsim10^{-32}$–$10^{-27}$ cm² as geometric saturation occurs and scattering departs from the Born regime; model-independent interpretation then ceases to be valid (Digman et al., 2019).

Quantum Corrections and Blind Spots

Loop-level corrections can dominate or cancel the tree-level amplitude, especially near parameter regions like Higgs resonance ($m_{\rm DM}\simeq m_h/2$ in IDM) or in pseudoscalar/Higgs-portal models with suppressed SI tree-level contributions (Abe et al., 2015, Azevedo et al., 2018). Theoretical uncertainties from matrix elements ($\lesssim30\%$) and twist-2 QCD running ($\lesssim20\%$) are subdominant here (Hisano et al., 2015). An irreducible quantum “floor” exists in several models (e.g., EW multiplets) (Hisano et al., 2011).

3. Experimental Determinations and Limits

Experimental efforts probe σ_χN over ≥90 orders of magnitude in DM mass and ≥25 orders in cross-section, via the following methodologies:

Direct Terrestrial Searches

Cryogenic and noble-liquid detectors (XENON1T, LUX, PandaX, DEAP-3600, CRESST, SENSEI, XQC, DAMIC, etc.) set upper bounds by searching for nuclear recoils due to DM scattering. Current SI upper limits reach $\sim10^{-47}$ cm² at $m_{\chi}\sim 50$ GeV (XENON1T), while SD bounds are weaker, $\sim10^{-41}$–$10^{-40}$ cm² (PICO, XENON1T) (Fushimi et al., 2021, Bramante et al., 2018, Mahdawi et al., 2018).

Null Heating Constraints and Earth-Captured Thermal Populations

“Anomalous heating” experiments—such as LN<sub\>2</sub> dewar boil-off measurements—constrain the possible density and cross-section of Earth-bound, thermalized DM with SI cross-sections $$\sigma_{\rm cr,\,300\,K} \leq 1.32\times10^{-27} n_{14}^{-1} (m_{\rm DM}/2 m_p)^{-1/2}$$ cm² at $$n_{14}\equiv n_{\rm DM}/(10^{14}\hspace{1mm}{\rm cm}^{-3})$$ (Neufeld et al., 2019). Combining with crust heat-flow arguments, these set stringent terrestrial limits for low-mass, strongly interacting DM—prohibiting $n_{\rm DM} \gtrsim 1.6\times10^{13}$ cm$^{-3}$ at Earth's surface.

Indirect Constraints via Astrophysical and Cosmological Observables

Rare kaon decays, Big Bang Nucleosynthesis (BBN), Cosmic Microwave Background (CMB) $N_\text{eff}$, and Lyman-α forest structure all provide limits on DM–nucleon interactions for sub-GeV DM. For example, $K^+\rightarrow\pi^+ +$ invisible decays via effective DM–quark/gluon couplings yield

$$\sigma^{\rm SI}_{\chi N} \lesssim 10^{-30}\hspace{1mm}\text{(gluon)},\,10^{-32}\hspace{1mm}\text{(quark)}\,\text{cm}^2$$

for $m_\chi\ll m_K/2$ (Cox et al., 22 Aug 2024). BBN/CMB coupling via 1-loop meson–photon diagrams closes the window for sub-MeV hadronically interacting DM with $\sigma_{\chi N} \gtrsim 10^{-29}$ cm² due to thermalization constraints.

Neutrino Experiments as Indirect Probes

Terrestrial and solar neutrino detectors (Super–Kamiokande, BOREXINO, SNO+, JUNO) constrain $\sigma_{\chi N}$ by searching for anomalous $\bar{\nu}_e$ or multiscatter events from captured DM annihilation in the Earth or Sun (Chan et al., 2022, Bramante et al., 2018). For WIMP masses $\sim20$–$50$ GeV, the 90% C.L. SI exclusion contours from Super–K neutrino fluxes can reach $10^{-43}$ cm², out-performing some direct-detection bounds in the same mass range.

Cosmic-ray Propagation and Inelastic Constraints

Inelastic DM-proton scattering, probed by cosmic-ray spallation (modifications in the Boron/carbon ratio measured by AMS-02 and DAMPE), can exclude $\sigma_{\chi p}^{\rm inel} \lesssim 10^{-32}$ cm² at $m_\chi \simeq 2$ MeV—an improvement of several orders of magnitude relative to direct-detection extrapolations for sub-GeV DM (Lu et al., 2023).

Stellar and Neutron Star Probes

DM–nucleon interactions alter stellar cooling, neutron star capture and heating, and stellar structure, probing suppressed cross-sections inaccessible to terrestrial experiments. For bosonic (scalar or vector) DM with momentum/velocity suppression, neutron star heating bounds can reach $\sigma_{\chi N} \lesssim 10^{-48}$–$10^{-50}$ cm² for $m_\chi\sim 1$–$10^3$ GeV (Maity et al., 2021).

4. Velocity and Momentum Dependence, Resonances, and Scaling Laws

The canonical assumption is $\sigma_{\chi N}$ is velocity-independent, but many scenarios yield velocity power-law or resonant behaviors:

• Velocity scaling: For long-range (e.g., milli-charged) DM, $\sigma_{\chi N}\propto v^{-4}$; for dipole DM, $v^{\pm 2}$ (Mahdawi et al., 2018).
• Momentum suppression: Pseudoscalar/Higgs-portal and pseudo-Nambu-Goldstone (pNG) scenarios exhibit leading couplings that vanish as $q\to 0$, yielding "blind spots" for direct-detection (Azevedo et al., 2018, Abe et al., 24 Nov 2024).
• Resonant scattering: Attractive Yukawa potentials produce quantum-mechanical s-wave and higher partial-wave resonances. Near resonance, $\sigma$ can deviate sharply from perturbative expectations and acquire strong, nontrivial $v$-dependence (Xu et al., 2021).
• Breakdown of A<sup\>4</sup> scaling: Geometric saturation occurs at $\sigma_{\chi N}\sim10^{-32}$–$10^{-27}$ cm², where cross sections for heavy nuclei such as Xe reach their geometric limit ($\pi r_A^2$), necessitating full nuclear modeling and invalidating naive scaling (Digman et al., 2019).

5. Upper, Lower, and Physical Bounds on Cross-Section Magnitudes

Physical, astrophysical, and model-structural considerations introduce strict upper and lower limits:

• Upper bound (pointlike DM): $\sigma_{\chi N}^{\rm max} \sim 10^{-25}$ cm². Beyond this, Born approximation fails; composite DM or geometric considerations become necessary, and cross-section scaling becomes model-dependent (Digman et al., 2019).
• Lower bound (terrestrial constraints): Anomalous heat-flow and local overburden impose lower bounds for strongly-interacting DM, e.g., $\sigma_{\rm cr,300K} \gtrsim 5 \times 10^{-29}$ cm² to avoid excessive heat transport through Earth's crust (Neufeld et al., 2019).
• Composite DM regime: For $\sigma_{\chi N} > 10^{-25}$ cm², only DM with substructure and large effective size can accommodate such cross sections, but the scaling with nucleon number is no longer universal (Digman et al., 2019).
• Unitarity and quantum limits: Unitarity bounds allow, on resonance, cross sections as large as $\sim4\pi/(\mu^2 v^2)\sim5\times10^{-21}$ cm² for nonrelativistic DM—showing that even larger cross sections do not imply compositeness per se (Xu et al., 2021).

6. Experimental Sensitivities and Model Interpretations

Scattering cross-section constraints yield critical model-exclusion regions:

• WIMP (electroweak multiplet) scenario: Predicts $\sigma_{\chi N}^{\rm SI} \sim 10^{-48}$–$10^{-46}$ cm², generically below current direct-detection limits but accessible to upcoming multi-ton detectors (Hisano et al., 2011, Hill et al., 2011, Fushimi et al., 2021).
• Inert Doublet or Higgs-portal scenarios: Loops can substantially affect cross sections, especially in “blind spot” regions or when $\lambda_{hH}$ is small; $\sigma_{\chi N}$ can range from $10^{-49}$ to above $10^{-46}$ cm² in the Higgs-pole region (Abe et al., 2015).
• Pseudo-Nambu-Goldstone DM: Momentum-suppressed cross sections as low as $10^{-60}$ cm² at tree level, but effective two-component scenarios can yield detectable rates via subdominant vector/scalar DM components (Abe et al., 24 Nov 2024).
• Terrestrial hidden-DM scenarios: Constraints on $\sigma_{\chi N}$ down to $10^{-27}$ cm² for $m_\chi\sim2m_p$ preclude large Earth-bound densities and exclude hadronic DM-capture scenarios leading to detectable anomalous heating (Neufeld et al., 2019).

Measured or projected cross-section sensitivities are summarized in the following representative table:

DM context	Best exclusion on σ<sub>χN</sub> (cm²)	Mass range	Source
LN<sub\>2</sub> boil-off heating	$1.3\times10^{-27}$	$\sim 1$–$2\,m_p$	(Neufeld et al., 2019)
Super-K neutrino anti-ν<sub>e</sub>	$1.0\times10^{-43}$	$20$–$50$ GeV	(Chan et al., 2022)
Cosmic-ray spallation (CR B/C)	$1\times10^{-32}$	$2$ MeV	(Lu et al., 2023)
Kaon decay (K→π+χχ, gluon DM)	$3\times10^{-30}$	$<100$ MeV	(Cox et al., 22 Aug 2024)
Direct-detection (XENON1T SI)	$4\times10^{-47}$	$40$–$50$ GeV	(Fushimi et al., 2021)
Neutron star heating (bosonic DM)	$1\times10^{-48}$	$1$–$10^3$ GeV	(Maity et al., 2021)
Geometric limit (Xe, model breakdown)	$1\times10^{-24}$	$>1$ GeV	(Digman et al., 2019)

7. Outlook: Model Dependence, Uncertainties, and Open Problems

Significant uncertainties in $\sigma_{\chi N}$ calculations arise from both hadronic input (scalar form factors, twist-2 matrix elements) and higher-order matching and running, with individual sources typically $\lesssim 10$–30% (Hisano et al., 2015, Hisano et al., 2011). However, in kinematic or parameter-region “blind spots,” quantum effects, non-perturbative resonances, or suppressed couplings, cross-section predictions can span orders of magnitude.

Breakdown of Born-level nuclear scaling, momentum/velocity suppression, and the requirement of composite or extended DM structure for large cross-section scenarios remain crucial challenges for universally interpreting experimental results in terms of σ_χN (Digman et al., 2019, Xu et al., 2021, Azevedo et al., 2018). Model-independent analyses must confront the non-universality of scaling at high cross-section, the limitations of current theoretical uncertainties, and the necessity for experimental sensitivity across diverse velocity and recoil ranges.

In summary, the dark-matter–nucleon scattering cross-section is a highly model-sensitive, scale- and kinematically-dependent parameter, central to all avenues of DM detection and phenomenology. Its determination requires rigorous field-theoretic computation, careful scaling interpretations, and a critical understanding of experimental and astrophysical limits across all relevant DM scenarios.