Inelastic WIMP-Nucleus Scattering

• Inelastic WIMP-nucleus scattering is a process where a dark matter particle collides with a nucleus, causing either the particle or nucleus to transition to an excited state.
• This mechanism involves kinematic thresholds that shift recoil energy distributions, critically affecting direct detection experiments.
• Advanced effective field theory methods and multi-signal discrimination techniques are used to differentiate inelastic events from elastic scattering and to constrain WIMP properties.

Inelastic WIMP-nucleus scattering refers to processes in which a weakly interacting massive particle (WIMP) undergoes a collision with a nucleus that does not leave the nucleus (or the WIMP) in its ground state. This contrasts with elastic scattering, where all particles remain in their initial quantum states except for recoil kinetic energy. Inelastic processes include two main classes: (a) WIMPs upscattering to an excited internal state (“inelastic dark matter,” or iDM), and (b) the nuclear target being excited to a low-lying state (nuclear inelastic channel). Both types of inelasticity have pronounced kinematic and nuclear-structure consequences, significantly impacting direct dark matter detection signatures and experimental strategies.

1. Theoretical Framework of Inelastic WIMP-Nucleus Scattering

The inelastic WIMP-nucleus scattering scenario arises when a dark matter particle χ interacts with either a nucleus N or itself in such a way that an internal energy splitting δ must be supplied or released. The canonical inelastic dark matter (iDM) model considers the process

$\chi + N \rightarrow \chi^* + N$

where $\chi^*$ is a state of mass $m_\chi^* = m_\chi + \delta$. The energy splitting $\delta$ is typically chosen to be on the order of the kinetic energy of halo WIMPs ($\mu v^2$, with μ the reduced WIMP–nucleus mass and v the velocity of the WIMP). This introduces a kinematic threshold: only sufficiently energetic WIMPs can upscatter, shifting the sensitive recoil energy range and altering the rate.

The minimum kinematically allowed velocity for such a transition is

$$v_{\min} = \sqrt{\frac{1}{2 m_N E_R}} \left( \frac{m_N E_R}{\mu} + \delta \right)$$

where $E_R$ is the recoil energy and $m_N$ the target mass (Chang et al., 2010). This formula reflects the requirement to provide both nuclear recoil and the excitation energy $\delta$. In the case where the nucleus is excited, the analogous kinematic considerations apply, but $\delta$ is replaced by the nuclear excitation energy $E^*$ (Baudis et al., 2013).

At the level of effective field theory (EFT), inelastic interactions may arise from higher-dimensional operators (operators $O_j$ with momentum or velocity dependence), and the most general Galilean-invariant Hamiltonian includes both spin-independent (SI) and spin-dependent (SD) contributions, as well as numerous possible velocity/momentum-dependent terms (Anand et al., 2014, Gondolo et al., 2020).

2. Nuclear Structure and Multipole Response

Nuclear structure profoundly affects the inelastic process. For nuclear excitations, the final state must be physically accessible: typically, only the lowest-lying nuclear excited states (10–100 keV) are relevant. The nuclear response to both SI and SD couplings is encoded in operator-specific structure factors $S(p)$ computed via shell-model or deformed-shell-model techniques, including contributions from chiral EFT-derived WIMP–nucleon currents (Baudis et al., 2013, Vietze et al., 2014, Sahu et al., 2020, Sahu et al., 2020). Only selected multipoles are allowed by angular momentum and parity constraints. For instance, in SI inelastic transitions involving odd-mass xenon isotopes, L=2 dominates, as the monopole L=0 is forbidden for transitions between nonidentical states (Vietze et al., 2014).

In the non-relativistic EFT framework,

$$\mathcal{H}(\mathbf{r}) = \sum_{\tau=0,1} \sum_j c_j^\tau \mathcal{O}_j(\mathbf{r}) t^\tau$$

with the $O_j$ spanning SI, SD, and velocity/momentum-dependent couplings (Gondolo et al., 2020). Nuclear response functions are obtained via multipole expansion, producing six independent nuclear response functions (not just SI/SD). For inelastic transitions, additional multipoles become non-vanishing. The nuclear structure functions $S_{ij}(u)$ (with $u = q^2b^2/2$) and their normalization via static spin factors $\Omega_{0,1}$ enable computation of differential and integrated rates (Vergados et al., 2015, Vergados et al., 2016, Sahu et al., 2020).

3. Kinematics, Parameter Space, and Experimental Implications

In inelastic upscattering, the extra energy $\delta$ (or $E^*$ for nuclear excitation) sharply raises the minimum WIMP velocity for scattering, causing a strong suppression of rates for light targets or for small $v_{\max}$ relative to the required $v_{\min}$. This kinematics can “turn off” sensitivity in some experiments, while leaving others (those using heavier nuclei) comparatively unaffected (Chang et al., 2010). For example, in the DAMA NaI(Tl) experiment, only thallium impurities (A ≈ 205) are kinematically accessible at large $\delta$, despite their low concentration (Chang et al., 2010). General parameter space analyses reveal “allowed islands” in $\delta$–$m_\chi$, e.g., $m_\chi \sim 100\,\rm GeV$, $\delta \sim 200\,\rm keV$. Such regions may be consistent with DAMA while escaping null results from CRESST-II, XENON100, or other lighter-target experiments (Chang et al., 2010, Kang et al., 2019).

In the case of nuclear inelastic channels, only WIMPs with velocities in the high-velocity tail of the halo distribution can induce excitation, leading to sharply peaked or cutoff spectra at characteristic recoil energies. Nuclear de-excitation yields prompt $\gamma$ emissions, yielding a combined electronic plus nuclear recoil signal (“multi-signal signatures”) in the detector (Baudis et al., 2013, Adhikari et al., 2023).

4. Identification Strategies and Model-Independent Analysis

Distinguishing inelastic from elastic scenarios and reconstructing WIMP properties requires model-independent approaches. One technique fits the differential recoil spectrum to a two-parameter exponential form,

$$\frac{dR}{dQ}\big|_{\rm exp} = r_0\,\exp\left(-kQ - \frac{k'}{Q}\right)$$

where $k'$ is zero in elastic scenarios but nonzero for inelastic scattering, and its value encodes sensitivity to $\delta$ (Miao et al., 2013). The spectrum exhibits a characteristic maximum at $Q_{\rm thres} = [m_\chi/(m_\chi + m_N)]\delta$. Measurement of this energy (preferably with different target nuclei) enables simultaneous, model-independent extraction of $m_\chi$ and $\delta$ even with modest statistics ($\mathcal{O}(50)$ events) (Miao et al., 2013). Even small deviations from the exponential spectrum expected for elastic scattering can be used to refute or confirm the iDM hypothesis.

Combined data from different targets—especially with substantially different masses—break degeneracies and provide robust constraints on both the inelastic mass splitting and the WIMP mass (Miao et al., 2013). By systematically marginalizing over the operator coefficients in a full NR EFT, one obtains joint constraint regions as intersections of multi-dimensional ellipsoids in operator-coupling space, sharply circumscribed in the $(m_\chi, \delta)$ plane (Kang et al., 2019).

5. Experimental Results and Limit-Setting

Multiple direct searches have implemented dedicated analysis channels for inelastic WIMP–nucleus scattering. For the nuclear inelastic channel, the unique “nuclear recoil plus prompt de-excitation gamma” signature is distinctive. Key results include:

• XMASS–I and XENON100, XENON1T have set upper limits on inelastic WIMP–nucleus cross sections using ${}^{129}$Xe and ${}^{127}$I targets, typically at the level of $\sim 10^{-39}$ cm$^2$ for $m_\chi \sim 100$–$200$ GeV (Uchida et al., 2014, Aprile et al., 2017, Collaboration et al., 2020, Collaboration et al., 2018, Adhikari et al., 2023).
• The COSINE-100 experiment exploited the 57.6 keV $\gamma$ tag from $^{127}$I excitation, setting limits on the spin-dependent WIMP–proton inelastic cross section at $1.2 \times 10^{-37}\,\mathrm{cm}^2$ for a 500 GeV WIMP (Adhikari et al., 2023).
• In the iDM upscattering scenario, the kinematic thresholds cause many experiments to see no signal for certain $(m_\chi, \delta)$ regions, reconciling DAMA’s modulation with null results elsewhere only at the cost of fine-tuned WIMP mass and $\delta$, and frequently only when the dominant scattering is off rare high-A impurities (e.g., thallium) (Chang et al., 2010).
• Advanced analysis methods exploit event-level discrimination: scintillation time profiles (to distinguish nuclear recoil/gamma from $\beta$ backgrounds), event topology (single-hit/multi-hit separation), and multi-dimensional likelihood fits incorporating both electron-equivalent and nuclear recoil charges (Collaboration et al., 2018, Collaboration et al., 2020).

A practical summary of published upper limits (for selected targets, exposure, and $m_\chi$) is shown in the table below.

Experiment (Target)	Channel	Upper limit on $\sigma_{\rm inelastic}$ [cm$^2$]	$m_\chi$ [GeV]
XMASS-I ($^{129}$Xe)	nuclear inelastic (NR+$\gamma$)	$3.2 \times 10^{-39}$	50–200
XENON1T ($^{129}$Xe)	nuclear inelastic (NR+$\gamma$)	$3.3 \times 10^{-39}$	130
COSINE-100 ($^{127}$I)	nuclear inelastic (NR+$\gamma$)	$1.2 \times 10^{-37}$	500
XMASS-I ($^{129}$Xe)	SD n inelastic	$4.1 \times 10^{-39}$	200

6. Nuclear Physics and Operator Discrimination

Nuclear response functions for inelastic transitions are generally much more suppressed for SI operators than SD: the SI inelastic structure factor is typically reduced by four orders of magnitude compared to the elastic channel due to the absence of coherent L=0 multipoles (Vietze et al., 2014). In contrast, the analogous suppression for SD inelastic scattering is only about an order of magnitude. Therefore, observation of an inelastic nuclear excitation at a rate comparable (within an order of magnitude) to the elastic one strongly favors an SD coupling. This provides a direct discriminator between SI and SD interactions (Vietze et al., 2014, Baudis et al., 2013, Vergados et al., 2015).

The effective field theory approach encapsulates this through the operator basis (e.g., $\mathcal{O}_4 = S_\chi \cdot S_N$ for SD, $\mathcal{O}_1$ for SI, higher $O_j$ for derivative/velocity-dependent). Inelastic transitions selectively amplify certain operators (notably those with higher multipolarity structure or velocity/momentum dependence), and their recoil spectra peak at higher energies with sharp upper and lower bounds (Arcadi et al., 2019).

7. Extended Signatures: Migdal Effect, Exothermic Channels, and Multicomponent Scenarios

The Migdal effect, whereby a WIMP-induced nuclear recoil causes ionization of the atomic electron shell, leads to the emission of detectable electrons or X-rays even when nuclear recoils are sub-threshold. In inelastic scenarios, this effect can enhance sensitivity to low-mass WIMPs (especially for exothermic, i.e., downscattering transitions with $\delta<0$), since the boost from the negative mass difference plus the EM energy from Migdal ionization shifts the signal into the observable energy region. For exothermic iDM, bounds from Migdal channels now dominate at low $m_\chi$ ($\lesssim$ 1 GeV), for instance, using XENON1T, DS50, or SuperCDMS data (Kang et al., 23 Jul 2024).

Models with pseudo-Nambu–Goldstone dark matter, for example, display derivative (momentum-suppressed) scattering, leading to “inelastic-like” behaviors: effective nuclear cross sections vanish at zero momentum transfer and dominate only for subleading dark-sector states, with striking consequences for direct detection exclusion curves (Abe et al., 24 Nov 2024).

8. Conclusion and Open Questions

Inelastic WIMP–nucleus scattering fundamentally broadens the range of experimental signatures in direct dark matter searches and alters how limits are derived, compared, and interpreted. It provides a critical context for resolving apparent inconsistencies between modulation signals in DAMA and null results elsewhere, though current analyses suggest only fine-tuned regions in parameter space remain viable (Chang et al., 2010, Kang et al., 2019). Model-independent EFT analyses and advanced detection methods (involving multi-signal discrimination, Migdal effect analysis, and exploitation of specific nuclear excited states) are essential for fully characterizing such scenarios (Miao et al., 2013, Tomar et al., 2021, Kang et al., 23 Jul 2024).

The interplay between kinematic thresholds (controlled by $\delta$ or $E^*$), halo velocity distributions, and detailed nuclear structure underpins current and future experimental strategies. The continued development of high-resolution detectors, background discrimination techniques, and multi-target campaigns—especially with heavy and odd nuclei—is expected to further constrain or potentially reveal inelastic WIMP–nucleus scattering in the next generation of dark matter searches.