Neutron DM-Nucleon SI/SD Interactions

• Spin-independent (SI) interactions couple to the nuclear mass density with an A² enhancement, while spin-dependent (SD) interactions couple to unpaired nucleon spins.
• Variations in neutron-proton coupling ratios and nuclear structure functions create significant uncertainties and distinct experimental signatures.
• Combined analyses of SI and SD channels, considering astrophysical and nuclear uncertainties, are vital for accurate dark matter detection and model discrimination.

Spin-independent (SI) and spin-dependent (SD) dark matter (DM)–nucleon interactions constitute the principal frameworks for the interpretation of constraints from direct, indirect, and collider dark matter searches. Neutron interactions are especially relevant for experiments sensitive to odd‑neutron nuclei, such as xenon-based detectors and solar capture constraints. SI interactions couple primarily to the mass density of the nucleus and exhibit an approximate $A^2$ enhancement for isospin-symmetric couplings. SD interactions instead couple to angular momentum carried by unpaired nucleons, with sensitivities governed by the admixture of proton and neutron spins in a given target. The interplay of SI and SD interactions, the neutron-proton coupling ratio, and the possibility of models with uniquely dominant neutron couplings or suppression of one channel relative to the other, all play essential roles in interpreting and combining current experimental results.

1. Theoretical Formalism of SI and SD Neutron Interactions

The SI and SD interactions are typically described within the nonrelativistic effective operator framework. For a spin-$\frac{1}{2}$ WIMP $\chi$ and nucleon $N$:

• SI operator: $\mathcal{O}_1 = \mathbb{1}_\chi \mathbb{1}_N$
• SD operator: $\mathcal{O}_4 = \mathbf{S}_\chi \cdot \mathbf{S}_N$

Each operator admits isoscalar ($\tau=0$) and isovector ($\tau=1$) couplings, with WIMP–neutron (or proton) couplings given by $c_{i,n}=c_i^0-c_i^1$ and $c_{i,p}=c_i^0+c_i^1$ (Bäckström et al., 2018).

The zero-momentum-transfer per-nucleon cross sections are

$$\sigma_n^{\rm SI} = \frac{\mu_n^2}{\pi}\,|c_{1,n}|^2,\qquad \sigma_n^{\rm SD} = \frac{3\mu_n^2}{\pi}\,|c_{4,n}|^2,$$

with $\mu_n$ the DM–neutron reduced mass (Peters et al., 2021, Bäckström et al., 2018).

For scattering on nuclei, the SI rate is coherently enhanced,

$$\sigma_{\rm SI}^N(0) = \frac{4}{\pi}\mu_N^2[Z f_p + (A-Z) f_n]^2,$$

with $f_n/f_p$ the neutron–to–proton coupling ratio, and $Z, A$ the atomic and mass numbers (Marcos et al., 2015). The SD cross section for a nucleus of spin $J$ is

$$\sigma_{\rm SD}^N(0) = \frac{32}{\pi}\mu_N^2 G_F^2 \frac{J+1}{J} [a_p \langle S_p\rangle + a_n \langle S_n \rangle]^2,$$

with $a_{n,p}$ the axial couplings and $\langle S_{n,p}\rangle$ the neutron/proton spin expectation values (Marcos et al., 2015).

2. Parameter Dependencies, Neutron-Proton Coupling Ratios, and Nuclear Structure

The sensitivity of both SI and SD channels to neutron couplings reflects key nuclear and particle physics properties.

• Isospin structure (SI): $f_n/f_p$ can deviate from unity—well-known isospin-violating scenarios include $f_n/f_p \approx -0.7$ (“Xe-phobic”), which suppresses the response in xenon targets by over an order of magnitude (Marcos et al., 2015). SI bounds are thus highly sensitive to the assumed neutron/proton ratio.
• Axial-vector structure (SD): $a_n/a_p$ sets whether the neutron or proton spin term dominates. Odd-neutron nuclei (e.g., $^{129}$Xe, $^{73}$Ge) provide leading sensitivity to $a_n$, whereas odd-$Z$ nuclei (e.g., $^{19}$F, $^{23}$Na) target $a_p$. Variation in $a_n/a_p$ alters exclusion domains in the $(\sigma_p^{\rm SI}, \sigma_p^{\rm SD})$ plane, sometimes creating interference and shifting limits by orders of magnitude at low masses (Marcos et al., 2015).
• Nuclear response functions: For SD scattering, the choice of nuclear structure calculations (e.g., chiral effective theory, Bonn A, Nijmegen, various shell-model inputs) can shift the SD–proton limit by up to a factor of 10 and SD–neutron by smaller but significant amounts. This introduces $O(1$–$10)$ uncertainties in SD limits (Marcos et al., 2015).

3. Combined Limit-Setting and Astrophysical/Nuclear Uncertainties

Significant advances in limit setting arise from considering both SI and SD interactions jointly and varying $f_n/f_p$ and $a_n/a_p$ freely.

• The expected event rate per recoil energy is

$$\frac{dR}{dE_R} = \frac{\rho_0}{m_\chi m_N}\int_{v_\text{min}}^{v_\text{esc}} v f(v)\left( \frac{d\sigma_{\rm SI}}{dE_R} + \frac{d\sigma_{\rm SD}}{dE_R} \right) dv,$$

and detection prospects hinge on the full phase space of couplings, form factors, and astrophysical velocity distributions (Marcos et al., 2015).

• For realistic analysis, SI and SD predicted events are expressed as

$$N_{\rm SI} = \sigma_p^{\rm SI}\left[ F_Z^p + (f_n/f_p)^2 F_Z^n + (f_n/f_p) F_Z^{pn}\right],$$

$$N_{\rm SD} = \sigma_p^{\rm SD}[C_{00}F_Z^{00} + C_{01}F_Z^{01} + C_{11}F_Z^{11}],$$

with $F_Z$ arrays and $C_{ij}$ fixed by the ratio $a_n/a_p$ (Marcos et al., 2015).

• Self-consistent Milky Way velocity distributions for NFW, Einasto, and Burkert profiles can change limits by $O(1)$ at $m_\chi \lesssim 30\,$GeV (Marcos et al., 2015).
• Neglecting SD interactions or assuming $f_n/f_p=1$ can incorrectly leave large allowed parameter regions that are otherwise excluded when both effects are included. The combined SI+SD limits expose complex, often “rectangular,” exclusion domains in the $\sigma^{\rm SI}$–$\sigma^{\rm SD}$ space (Marcos et al., 2015).

4. Phenomenology in Specific DM Scenarios

In the MSSM, DM–neutron SI and SD cross sections are tightly correlated except in finely tuned or loop-induced scenarios.

• Tree-level generation: SI arises primarily via CP-even Higgs exchange, and SD via $Z^0$ exchange, with per-nucleon effective couplings

$$\sigma_{\rm SI}^n \sim \frac{4}{\pi} \mu_n^2 f_n^2,\qquad \sigma_{\rm SD}^n \sim \frac{12}{\pi} \mu_n^2 a_n^2,$$

where $f_n$ (Higgs-induced) and $a_n$ (axial $Z$-exchange) coefficients depend on the neutralino composition (Cohen et al., 2010).

• Correlations: In the absence of numerical conspiracies (“well-tempered” neutralinos), sizable SD cross sections imply SI cross sections at or above the near-future sensitivity limits:

$$\frac{\sigma_{\rm SD}^p}{\sigma_{\rm SI}^p} \sim 8 \times 10^4 \left( \frac{|Z_{H_d}|^2 - |Z_{H_u}|^2}{(Z_W - t_w Z_B) Z_{H_u}} \right)^2.$$

For typical mixing parameters $\sim0.1$, the ratio falls in the $10^4$–$10^5$ range (Cohen et al., 2010).

• Exceptions: Only special cases, e.g., accidental blind spots (equality of Higgsino admixtures), isospin-violating Higgs couplings, or models with light pseudoscalar mediators, yield detectable SD with negligible SI. For example, a light pseudoscalar mediator ($\phi$) gives tree-level neutron SD-only coupling,

$$\sigma_{\rm SD}^n \sim \frac{3}{\pi} \mu_{\chi n}^2 \left( \frac{g_\chi g_n}{m_\phi^2} \Delta_n \right)^2,$$

with $\Delta_n$ the neutron spin fraction, while the SI component arises only at suppressed loop level ($\sigma_{\rm SI}/\sigma_{\rm SD} \sim 10^{-8}$–$10^{-12}$), rendering SI practically unobservable in this case (Freytsis et al., 2010).

5. Experimental Constraints: Direct, Indirect, and Ceiling Effects

Direct-detection and neutron-specific reach

• Xe, Ge-based recoil detection: With large $\langle S_n \rangle$ in xenon and germanium targets, neutron-coupled SD interactions dominate the accessible SD parameter space for these detectors (Marcos et al., 2015). Limits depend sharply on $a_n$ and associated structure factors.
• Surface/low-threshold experiments: For SD DM–neutron and SI channels, sensitivity is ultimately bounded at high cross section by "ceiling" effects—DM particles scatter in the atmosphere and Earth before reaching the detector (Collaboration et al., 14 Feb 2025). For example, the QUEST-DMC superfluid helium-3 experiment obtains

$$\sigma_{\chi n}^{\rm SD,\,ceil}\sim 3\times 10^{-24}\,\mathrm{cm}^2,$$

$$\sigma_{\chi N}^{\rm SI,\,ceil}\sim 7.5\times 10^{-27}\,\mathrm{cm}^2$$

for sub-GeV masses, using a diffusion model for atmospheric propagation (Collaboration et al., 14 Feb 2025).

• Neutron-multiplicity searches: Neutron production in heavy targets (e.g., Pb in NMDS-II) provides indirect constraints, with cross-section limits

$$\sigma_n^{\rm SI} \lesssim 10^{-45}\,\mathrm{cm}^2,\qquad \sigma_n^{\rm SD} \lesssim 2\times 10^{-42}\,\mathrm{cm}^2$$

over $300\,\mathrm{MeV}$–$100\,\mathrm{GeV}$ DM mass (Cao et al., 27 Jun 2025). Scaling depends on $A$ (SI: $A^{-2}$, SD: $A^{-1}$).

Neutrino telescopes and solar capture

• IceCube/PINGU: Constraints on WIMP capture by the Sun and subsequent neutrino annihilation signals limit neutron SI/SD couplings. The 90% C.L. bounds are

$$\sigma_{\rm SI}^n \lesssim 3\times10^{-43}\,\mathrm{cm}^2,\qquad \sigma_{\rm SD}^n \lesssim 5\times10^{-40}\,\mathrm{cm}^2$$

at $M_\chi=100\,$GeV (Peters et al., 2021, Bäckström et al., 2018). PINGU will improve low-mass reach, particularly for models yielding soft ($b\bar b$) annihilation channels (Bäckström et al., 2018).

Higher-order corrections and global fits

• NLO QCD corrections to neutralino–neutron SI/SD cross sections shift predictions by $30$–$40\%$, comparable to uncertainties from the nuclear matrix elements. Neglecting these can misestimate DM masses or compositions in global fits (Klasen et al., 2017).

6. Model-Building Implications and Sequestered SI/SD Phenomenology

A generic finding is that, except in models with accidental cancellations or with mediators enforcing selection rules, SI and SD cross sections are correlated; models with measurable SD neutron couplings almost always induce SI interactions above the neutrino floor unless they invoke special mechanisms.

Models where light pseudoscalar exchange dominates achieve uniquely SD-dominant scattering, as the leading SI channel appears only at one loop (box diagrams with scalar insertions), yielding

$$\sigma_{\rm SI}^n / \sigma_{\rm SD}^n \sim 10^{-8}\text{–}10^{-12},$$

well below experimental sensitivity thresholds. Thus, in this case, SD searches (especially with neutron‑sensitive targets) can provide the first and only direct-detection signal (Freytsis et al., 2010).

A plausible implication is that a detection in a neutron-SD dedicated experiment, in the absence of an SI signal in ton-scale detectors, would strongly suggest a light pseudoscalar mediator or similar mechanism enforcing SI suppression.

7. Summary Table: SI/SD DM–Neutron Coupling Features and Constraints

Aspect	SI (spin-independent)	SD (spin-dependent)
Operator	$\mathcal{O}_1=\mathbb{1}$	$\mathcal{O}_4=\mathbf{S}_\chi\cdot\mathbf{S}_N$
Nuclear scaling	$A^2$, isospin-dependent	$\sim (J+1)/J\, \langle S_n\rangle^2$
Typical coupling ratio impact	$f_n/f_p$ can suppress in Xe	$a_n/a_p$ sets neutron dominance
Model correlation	Tight, unless tuned or loop-only	Often tied to SI, except ps-scalar
Representative limit (Xe SD)	$\lesssim 10^{-45}~\mathrm{cm}^2$ (Cao et al., 27 Jun 2025)	$\lesssim 2\times10^{-42}~\mathrm{cm}^2$ (Cao et al., 27 Jun 2025)
Exceptional SI suppression	Only in loop‑induced (ps-scalar)	Pure SD, $\sigma_\mathrm{SI}\ll\sigma_\mathrm{SD}$ (Freytsis et al., 2010)

The SI and SD DM–neutron interaction landscape is characterized by model-dependent and nuclear-physics-dependent structure, strong interplay in the direct-detection interpretation, and the emergence of neutron SD as a powerful yet model-discriminating probe in certain scenarios. Combined SI and SD analyses, variation of neutron/proton coupling ratios, and careful treatment of nuclear/astrophysical uncertainties are now mandatory for a theoretically consistent extraction of dark matter constraints (Marcos et al., 2015, Klasen et al., 2017, Bäckström et al., 2018, Peters et al., 2021, Collaboration et al., 14 Feb 2025, Cao et al., 27 Jun 2025, Cohen et al., 2010, Freytsis et al., 2010).