Quantum-mechanical numerical model of interaction between dark atom and nucleus of substance

TL;DR

The paper tackles the three-body problem of a dark-atom bound state $X^{--}$–$^4He$ (XHe) interacting with a heavy nucleus by developing a quantum-mechanical numerical framework. It solves the isolated $OHe$ system to obtain ground and excited helium states, then extends to the $OHe$–nucleus system to compute the polarization via the Stark effect and the resulting dipole moment, and reconstructs the total effective potential including Coulomb, nuclear (Woods-Saxon), and centrifugal contributions. By combining these elements, it reveals a dipole Coulomb barrier that prevents fusion with ordinary nuclei and identifies a shallow bound region, providing a consistent mechanism for the stability of $OHe$ dark atoms and implications for direct-detection phenomenology. The approach lays groundwork for more precise modeling with finite-size effects and nuclear deformation, contributing to the viability assessment of composite dark matter scenarios in detector environments.

Abstract

The hypothesis of composite $XHe$ dark atoms may provide solution to the long-standing problem of direct searches for dark matter particles. The main problem of the $XHe$ dark atom is its ability to strongly interact with the nucleus of substance, arising from the unshielded nuclear attraction between the helium nucleus and the nucleus of matter. It is assumed that in order to prevent the destruction of the bound structure of dark atom, the effective potential of interaction between $XHe$ and the nucleus of substance must have dipole Coulomb barrier that prevents the fusion of dark matter atom particles with the nucleus of substance. The problem in describing the interaction between dark atom and substance nucleus is the three-body problem, for which an exact analytical solution is not available. Consequently, to assess the physical meaning of the proposed scenario, it is essential to develop a numerical approach. Our approach involves consistently developing an accurate quantum mechanical description of this three-body system, comprising bound dark atom and the external nucleus of substance. We incorporate the necessary effects and interactions to enhance the precision of the results, which helps to elucidate the most significant aspects of the proposed dark atom scenario.

Figures (4)

• Figure 1: Potentials of Coulomb (red dotted line), nuclear (green dotted line) and centrifugal (green solid line) interaction between helium and the nucleus of $Na$, the potential of Coulomb interaction between helium and $O^{--}$ particle (black dotted line) and the total interaction potential of the helium nucleus (blue dotted line) in the $O$He--$Na$ system at fixed $\Vec{R}_{OA}$. The red circle marks the value of the radius of the $He$ nucleus. Original authors’ figure taken from Bikbaev_2024.
• Figure 2: The total potential of helium in the $O$He--$Na$ system for fixed position of sodium $\Vec{R}_{OA}$ (blue solid line), graph of the squared modulus of the wave function of the ground state of helium in polarized dark atom for fixed $\Vec{R}_{OA}$ (red solid line), the intersection points of the graph of the total potential of helium and the graph of the squared modulus of the wave function of the ground state of helium (black circles). Original authors’ figure taken from Bikbaev_2024.
• Figure 3: Graph of the dependence of the dipole moment of polarized $O$He atom (red stars) on the radius vector of the outer sodium nucleus. Original authors’ figure taken from Bikbaev_2024.
• Figure 4: Graphs of Woods--Saxon nuclear potential (green dotted line), $U^{e}_{XHe}$ (blue dotted line), Stark potential (gray dotted line), centrifugal potential (purple dotted line) and total effective interaction potential of $O$He with the nucleus of the sodium (red dotted line) on the distance between the $He$ nucleus, located in the Bohr orbit of the $O$He atom, and the $Na$ nucleus for $J_{(OHe-Na)} = 3$. Original authors’ figure taken from Bikbaev_2024.