Inelastic Majorana Dark Matter Model

• The inelastic Majorana dark matter model describes a Dirac fermion receiving a small Majorana mass, resulting in two nearly degenerate states that interact via a light dark photon.
• It employs a detailed Lagrangian framework with kinetic mixing and Yukawa interactions to generate keV-scale mass splittings that explain the XENON1T electron-recoil signal and achieve the observed relic abundance.
• The model’s phenomenology is constrained by direct-detection thresholds and astrophysical requirements, with velocity-dependent self-interactions from dark photon exchange addressing small-scale structure issues.

The inelastic Majorana dark matter (iDM) model is a framework in which a fermionic dark sector, initially realized as a Dirac fermion, acquires a small Majorana mass splitting. This results in two nearly degenerate Majorana mass eigenstates. The dynamics are typically governed by interactions with light vector mediators—often referred to as dark photons—with communication to the Standard Model (SM) via kinetic mixing. These models have garnered significant interest for their ability to simultaneously address cosmological structure anomalies, explain laboratory excesses such as the XENON1T electron-recoil signal, and evade stringent direct-detection and collider constraints through kinematic thresholds and suppressed couplings.

1. Theoretical Construction and Field Content

The canonical Lagrangian for electroweak-scale iDM includes a Dirac fermion $\chi$ charged under a hidden $U(1)_X$ symmetry, acquiring both a Dirac mass $m_\chi$ and a small Majorana mass $\delta/2$ via a Yukawa interaction with a dark Higgs. After spontaneous symmetry breaking and diagonalization, the physical spectrum consists of two Majorana fields $\chi_1$ and $\chi_2$: $$m_{\chi_1} = m_\chi - \delta/2 \,,\qquad m_{\chi_2} = m_\chi + \delta/2\,,\qquad \delta \equiv m_{\chi_2} - m_{\chi_1} \,.$$ The dark vector boson $Z'$ (the dark photon), with mass $m_{Z'} \ll m_\chi$, mediates both self-interactions and inelastic transitions. The gauge kinetic mixing is parameterized by $\epsilon \ll 1$, coupling $Z'$ to the SM photon. The dark-sector Lagrangian in the mass basis relevant for these processes is

$$\mathcal{L}_{\rm DS} =\frac12\,\overline{\chi_1}(i\!\not\!\partial-m_{\chi_1})\chi_1 + \frac12\,\overline{\chi_2}(i\!\not\!\partial-m_{\chi_2})\chi_2 - \frac14\,Z'_{\mu\nu}Z'^{\mu\nu} +\frac{\epsilon}{2}\,Z'_{\mu\nu}F^{\mu\nu} - i\,g_X\,Z'_\mu\,\overline{\chi_2}\gamma^\mu \chi_1 \,.$$

Here, $g_X$ is the $U(1)_X$ gauge coupling, and $\alpha_X \equiv g_X^2/(4\pi)$ (Baek, 2021).

2. Origin and Phenomenological Role of Mass Splitting

The mass splitting $\delta$ arises from a UV-complete Dirac-Majorana seesaw structure, with the explicit breaking of the $U(1)_X$ symmetry via a small Majorana mass term for $\chi$. Diagonalization yields the two Majorana states $\chi_1, \chi_2$ with $m_{\chi_2} - m_{\chi_1} = \delta$. This splitting is a crucial parameter:

• For the XENON1T anomaly, $\delta \simeq 2.8~\mathrm{keV}$ yields the correct deposited electron recoil energy.
• For direct-detection constraints, $\delta$ determines the kinematic threshold for elastic and inelastic scattering processes and can forbid nucleon up-scatters at standard halo velocities (Baek, 2021).

3. Dark Matter Self-Interactions and Structure Formation

In the presence of a very light mediator ($m_{Z'} \ll m_\chi$), t-channel $Z'$ exchange produces a velocity-dependent self-interaction cross-section: $$\sigma_{\rm self} \approx \frac{4\pi\,\alpha_X^2\,m_\chi^2}{m_{Z'}^4}[1+\mathcal{O}(v^2)]\,.$$ Non-perturbatively, the cross-section must be computed by solving the Schrödinger equation with a Yukawa potential. The model can realize $\sigma/m_\chi \sim 1~\mathrm{cm}^2/\mathrm{g}$ at dwarf-galaxy velocities ($v \sim 30~\mathrm{km/s}$), with automatic suppression at cluster scales ($v \sim 1000~\mathrm{km/s}$), thereby addressing core–cusp and too–big–to–fail problems (Baek, 2021, Alvarez et al., 2019).

4. Inelastic Transitions, Direct Detection, and Laboratory Signatures

The process $\chi_1\chi_1 \to \chi_2\chi_2$ proceeds via the dark photon and has a threshold velocity: $v_{\min} = \sqrt{\frac{2\delta}{m_1}}\,.$ For $\delta \sim \mathrm{keV}$ and $m_\chi \sim 100~\mathrm{GeV}$, only the high-velocity tail of the halo can up-scatter. The prompt decay $\chi_2 \to \chi_1 Z'$ releases a $Z'$ with energy $\sim \delta$, leading to observable signatures:

• Inelastic up-scatter followed by $Z'$ absorption in xenon produces a mono-energetic electron-recoil spectrum. The absorption cross-section for a nonrelativistic $Z'$ is $$\sigma_{\mathrm{PE}}(E_{Z'}) = \epsilon^2\,\sigma_\gamma(E_{Z'})$$, where $\sigma_\gamma(E)$ is the SM photo-electric cross section (Baek, 2021).
• The XENON1T electron-recoil excess at $2.8~\mathrm{keV}$ is explained by $\delta = 2.8~\mathrm{keV}$, $\epsilon \sim 10^{-10}$, and $\sigma_{\rm inel} \sim 10^{-14}~\text{pb}$, reproducing the event rate (Baek, 2021, Dutta et al., 2021).

Direct detection via nucleon recoils is suppressed. Elastic $\chi_1$–nucleus scattering requires momentum transfer sufficient to bridge $\delta$, which is kinematically forbidden for typical WIMP velocities at keV-scale mass splittings, and the rate is further suppressed by $\epsilon^2$ (Baek, 2021).

5. Relic Density and Thermal History

The dominant freeze-out annihilation channel is $\chi_1\chi_1 \to Z'Z'$: $$\langle\sigma v\rangle_{\chi_i\chi_i\to Z'Z'} \simeq \frac{\pi\alpha_X^2}{m_\chi^2}\,,$$ with $\alpha_X \sim 3 \times 10^{-3}$ for $m_\chi \sim 100~\mathrm{GeV}$ yielding the observed DM relic abundance $\Omega_{\rm DM}h^2 \simeq 0.12$ (Baek, 2021). Hybrid freeze-in/freeze-out scenarios are also possible in light mediator regimes with additional singlet injection (Dutta et al., 2021).

6. Combined Phenomenological Constraints and Experimental Probes

A concise table illustrates the dependence of the key observables and constraints:

Observable/Constraint	Model Parameter(s)	Value/Threshold
Small-scale structure	$\sigma_{\rm self}/m_\chi$	$\sim 1~\mathrm{cm}^2/\mathrm{g}$ at $v\sim30~\mathrm{km/s}$
XENON1T electron recoil	$\delta$, $\epsilon$, $\sigma_{\rm inel}$	$2.8~\mathrm{keV}$, $10^{-10}$, $10^{-14}~\mathrm{pb}$
Relic abundance	$\alpha_X$	$\sim 3\times 10^{-3}$
Direct detection (nucleon)	$\delta$	Kinematically forbidden (keV gap)
Cluster bound	$\sigma/m_\chi$	$\lesssim 0.5~\mathrm{cm}^2/\mathrm{g}$ at $v\sim1000~\mathrm{km/s}$

The model's five parameters $(m_\chi,\delta,m_{Z'},\alpha_X,\epsilon)$ are tightly constrained yet consistent with all known data. Expanded frameworks with alternative mediators (scalar portals), different freeze-out mechanisms, or nonminimal gauge sectors (e.g., $U(1)_{L_\mu-L_\tau}$) can accommodate similar phenomenology and may address ancillary anomalies such as $(g-2)_\mu$ (Yang, 5 Dec 2025, Voronchikhin et al., 7 May 2025, Garcia, 4 Nov 2024).

Further parameter space is being scrutinized by next-generation direct-detection (e.g., DARWIN), fixed-target, and collider experiments (NA64, Belle II), particularly for sub-GeV dark matter and light mediators (Garcia et al., 13 May 2024, Voronchikhin et al., 7 May 2025). Astrophysical probes provide additional constraints based on density-core stability and core-collapse timescales in dwarfs; for low $m_\chi$ the mass splitting must exceed the up-scatter threshold to suppress halo dissipation (Alvarez et al., 2019).

7. Summary and Outlook

The inelastic Majorana dark matter paradigm connects small-scale structure solutions, laboratory anomalies, and cosmological abundance through a simple extension of the minimal hidden $U(1)$ sector. A characteristic feature is the presence of a keV–MeV mass splitting, light mediators (often with $m_{Z'} \lesssim 10^{-3}$–$10^{-1}$~eV), and kinetic mixing parameter $\epsilon \sim 10^{-10}$–$10^{-8}$, producing suppressed yet detectable signatures in deep-underground and accelerator-based experiments, while satisfying the relic density and evading ultrahigh-sensitivity direct-detection bounds (Baek, 2021, Dutta et al., 2021, Alvarez et al., 2019). Future searches will further test these models, with critical sensitivity in the electron-recoil channel, the sub-GeV DM regime, and halo structure observations.