Direct detection of right-handed fermionic dark matter: electron-recoil measurements in Xe atoms

Technical Summary: Direct Detection of Right-Handed Fermionic Dark Matter via Electron Recoil in Xenon

Problem and Motivation
The paper addresses the direct detection of a specific class of dark matter (DM) candidates: right-handed neutral fermionic particles (RHDMF) with masses in the sub-MeV range, specifically $O(10\text{--}200 \text{ keV}/c^2)$. While Weakly Interacting Massive Particles (WIMPs) in the GeV scale have been extensively searched for without success, and Warm Dark Matter (WDM) candidates (sterile neutrinos) in the keV range are studied for their cosmological implications, this work focuses on a distinct light dark matter (LDM) scenario. Motivated by recent astrophysical constraints on fermionic DM halos and the lack of positive WIMP detections, the authors investigate the interaction of these RHDMF particles with bound electrons in xenon atoms. Unlike absorption processes where the incident dark matter particle deposits both its rest mass energy and kinetic energy into bound atomic electrons, this study considers a scattering process where only kinetic energy is transferred.

Methodology and Interaction Model
The authors employ an effective field theory framework based on the infrared fixed point of a Nambu-Jona-Lasinio (NJL) type four-fermion interaction. The core mechanism involves the coupling of the right-handed DM fermion ($N_R$) to right-handed charged leptons via the $W$ boson in the infrared regime. This interaction allows the RHDMF to couple to Standard Model (SM) particles without requiring extra fields.

The specific process modeled is the inelastic scattering $\chi + e^- \to \nu + e^-$, where a right-handed DM fermion interacts with a bound electron in a xenon atom, producing a left-handed SM neutrino and an ionized electron. The interaction proceeds via a one-loop electromagnetic channel involving a virtual $W$ boson and a chiral flip. The effective Lagrangian includes a vertex where the RHDMF decays radiatively into a neutrino and a photon, which is then absorbed by the bound electron.

Key theoretical components include:

• Matrix Element: The squared matrix element $|\mathcal{M}|^2$ is derived for the one-loop process. It is expanded in powers of the DM velocity $v$. Given the non-relativistic nature of the DM ($v \sim 10^{-3}c$), higher-order velocity terms are negligible, and the calculation is dominated by the $B_0$ term, which depends on the recoil energy $E_R$ and the squared four-momentum transfer $q^2$.
• Kinematics: The authors derive a linear relationship between the momentum transfer $q$ and the recoil energy $E_R$ under the approximation $m_\chi v \ll q$, yielding $q = m_\chi + E_{nl}^B - E_R$, where $E_{nl}^B$ is the binding energy of the electron in shell $(n,l)$.
• Atomic Physics: The calculation explicitly incorporates the bound nature of the target electron. The event rate is modulated by an ionization form factor, $|f_{ion}^{nl}(k', q)|^2$, which accounts for the overlap between the initial bound state (described by Roothaan-Hartree-Fock wave functions) and the final continuum state (described by hydrogen-like wave functions with an effective charge).
• Event Rate Calculation: The differential event rate is computed for liquid xenon detectors, summing contributions from all occupied atomic shells. The theoretical recoil energy is mapped to the "visible" recoil energy using the relative efficiency of the XENONnT experiment.

Key Contributions

1. Generalization of Interaction Channels: The work extends previous results on effective electromagnetic channels by explicitly including the ionization form factor arising from the interaction of fermionic candidates with bound electrons, rather than treating electrons as free particles.
2. Specific Mass Range Analysis: The study focuses on the $25\text{--}200 \text{ keV}/c^2$ mass range, a region motivated by fermionic DM halo astrophysics but distinct from traditional WIMP or sterile neutrino searches.
3. Xenon Target Modeling: The authors provide a detailed calculation of event rates for xenon, utilizing specific radial wave functions and binding energies for various atomic shells (e.g., 1s, 2p, 3d, 4p) to model the ionization process accurately.
4. Exclusion Limits: Using the null results from the XENONnT experiment (1.16 ton-years exposure), the authors derive exclusion regions for the coupling constant $G_R$ and the total cross-section $\sigma_{LAB}$ as a function of the DM mass $m_\chi$.

Results

• Event Rates: The predicted event rates exhibit a strong dependence on the DM mass and the atomic shell. The spectra show peaks corresponding to ionization form factor enhancements at specific energies (e.g., $\sim 3, 8, 15, 30 \text{ keV}$). The authors note that for small visible recoil energies, the contribution is dominated by innermost shells due to a cascade effect where the outgoing electron ionizes outer shells.
• Exclusion Regions: The analysis sets upper limits on the coupling constant $G_R$ and the cross-section $\sigma_{LAB}$. The exclusion limits are most stringent for DM masses around $50 \text{ keV}/c^2$, where the predicted event rates are highest relative to the background. For this mass range, cross-sections above $\sim 10^{-46} \text{ cm}^2$ are excluded.
• Coupling Constraints: The derived constraints on $G_R$ are compared with cosmological limits (based on the age of the Universe). While the cosmological bound is generally tighter, the direct detection limits provide independent constraints that could improve with future experimental sensitivity.

Significance and Claims
The paper claims that the interaction of RHDMF with xenon electrons can be modeled as a three-step process involving virtual $W$ bosons and photons, leading to a distinct spectral signature in direct detection experiments. The authors emphasize that the mass-dependent peak structure in the predicted event rates, driven by the ionization form factor, offers a potential handle for distinguishing this signal from background in future experiments.

The work concludes that while no positive detection has been made, the current XENONnT data allows for the exclusion of specific regions in the parameter space, particularly around the $50 \text{ keV}/c^2$ mass scale. The authors assert that their results are consistent with other works utilizing similar mass ranges and that the specific shape of the exclusion curves (a "valley" around 50 keV) reflects the underlying physics of the ionization form factor. They suggest that if positive non-background events were observed in the future, the peak structure in the predicted spectrum could be used to determine the mass of the incoming particle, highlighting the importance of precise quantum atomic physics in future direct detection prospects.