The lack of positive evidence for Weakly Interacting Massive Particles (WIMPs) as well as the lack of discovery of supersymmetric (SUSY) particles at the LHC may appeal to a non-supersymmetric solution for the Standard Model problem of the Higgs boson mass divergence, the origin of the electroweak energy scale and the physical nature of the cosmological dark matter in the approach of composite Higgs boson. If the Higgs boson consists of charged constituents, their binding can lead to stable particles with electroweak charges. Such particles can take part in sphaleron transitions in the early Universe, which balance their excess with baryon asymmetry. Constraints on exotic charged species leave only stable particles with charge $-2n$ possible, which can bind with n nuclei of primordial helium in neutral dark atoms. The predicted ratio of densities of dark atoms and baryonic matter determines the condition for dark atoms to dominate in the cosmological dark matter. To satisfy this condition of the dark-atom nature of the observed dark matter, the mass of new stable $-2n$ charged particles should be within reach of the LHC for their searches. We discuss the possibilities of dark-atom binding in multi-atom systems and present state-of-the-art quantum mechanical descriptions of dark-atom interactions with nuclei. Annual modulations in such interactions with nuclei of underground detectors can explain the positive results of DAMA/NaI and DAMA/LIBRA experiments and the negative results of the underground WIMP searches.

The puzzle of the direct dark matter search can be resolved by examining the concept of «dark atoms», which consist of hypothetical stable lepton-like particles with a charge of $-2n$, where n is any natural number, bound to n nuclei of primordial helium. These «dark atoms», known as «XHe» (X-helium) atoms, remain undiscovered in experiments due to their neutral atom-like states. In this model, the positive results of the $DAMA/NaI$ and $DAMA/LIBRA$ experiments could be explained by the annual modulation of radiative capture of XHe atoms engaging in low-energy bound states with sodium nuclei. This specific phenomenon does not occur under the conditions of other underground experiments. The proposed solution to this puzzle involves establishing the existence of a low-energy bound state of «dark atoms» and nuclei while also considering the self-consistent influence of nuclear attraction and Coulomb repulsion. Resolving this complex issue, which has remained unsolved for the past 17 years, necessitates a systematic approach. To tackle this problem, numerical modeling is employed to uncover the fundamental processes behind the interaction of «dark atoms» with nuclei. To comprehend the essence of XHe’s interaction with baryonic matter nuclei, a classical model is employed wherein quantum physics and nuclear size effects are progressively incorporated. A numerical model describing the interaction between XHe «dark atoms» and nuclei is developed through the continuous inclusion of realistic features of quantum mechanics in the initial classical three-body problem involving the X-particle, the helium nucleus, and the target nucleus. This approach yields a comprehensive numerical model that encompasses nuclear attraction and electromagnetic interaction between the «dark atom» and nuclei. Finally, this model aids in supporting the interpretation of the results obtained from direct underground dark matter experiments through the lens of the «dark atom» hypothesis.

Within the framework of the XHe hypothesis, the positive results of the DAMA/NaI and DAMA/LIBRA experiments on the direct search for dark matter particles can be explained by the annual modulation of the radiative capture of dark atoms into low-energy bound states with sodium nuclei. Since this effect is not observed in other underground WIMP (weakly interacting massive particle) search experiments, it is necessary to explain these results by investigating the possibility of the existence of low-energy bound states between dark atoms and the nuclei of matter. Numerical modeling is used to solve this problem, since the study of the XHe–nucleus system is a three-body problem and leaves no possibility of an analytical solution. To understand the key properties and patterns underlying the interaction of dark atoms with the nuclei of baryonic matter, we develop the quantum mechanical description of such an interaction. In the numerical quantum mechanical model presented, takes into account the effects of quantum physics, self-consistent electromagnetic interaction, and nuclear attraction. This approach allows us to obtain a numerical model of the interaction between the dark atom and the nucleus of matter and interpret the results of direct experiments on the underground search for dark matter, within the framework of the dark atom hypothesis. Thus, in this paper, for the first time, steps are taken towards a consistent quantum mechanical description of the interaction of dark atoms, with unshielded nuclear attraction, with the nuclei of atoms of matter. The total effective interaction potential of the OHe–Na system has therefore been restored, the shape of which allows for the preservation of the integrity and stability of the dark atom, which is an essential requirement for confirming the validity of the OHe hypothesis.

The puzzle of direct dark matter searches can b solved in the model of an OHe dark atom, which consists of a stable $O^{--}$ lepton core and a nuclear interacting (alpha particle) shell of a primordial helium nuclei. In this model, positive results of the DAMA group can be explained by annual modulation of radiative capture of OHe atoms to low-energy bound states with sodium nuclei, which does not take place under the conditions of other underground experiments. The existence of such a low-energy bound state is the key problem of the OHe model of composite dark matter. The complexity of this problem, which has not found a correct solution during the last 15 years, requires a consistent approach to its solution. Within the framework of the proposed approach to such modeling, in order to reveal the essence of the processes of interaction of OHe with the nuclei of baryonic matter, a classical model is used, to which the effects of quantum physics and final size of nuclei are successively added. The numerical model of the interaction of the “dark” OHe atom with the nuclei is developed by successive addition of realistic features of a quantum-mechanical description to the initial classical problem of three point-like bodies ($O^{--}$ particle, the He nucleus and the target nucleus). The developed approach leads to a numerical model describing the OHe-nucleus system with self-consistent accounting for nuclear attraction and electromagnetic interaction of dark atom with nuclei. The model can prove the interpretation of the results of the direct underground experimental dark matter search in the terms of the dark atom hypothesis.

The hypothesis of composite $XHe$ dark atoms offers a compelling framework to address the challenges in direct dark matter particles detection, as their neutral, atom-like configuration evades conventional experimental signatures. A critical issue may arise in interaction between $XHe$ and atomic nuclei due to the unshielded nuclear attraction, which could destabilize the dark atom's bound state. To resolve this, we propose a novel numerical quantum mechanical approach that accounts for self-consistent electromagnetic-nuclear couplings. This method addresses to eliminate the inherent complexity of the $XHe$-nucleus three-body system, where analytical solutions are intractable. By reconstructing the effective interaction potential - including dipole Coulomb barrier and shallow potential well - we demonstrate how these features lead to the formation of $XHe$-nucleus bound states and modulate low-energy capture processes. Our model enables validation of the dark atom hypothesis, particularly in interpreting experimental anomalies like annual modulation signals observed in DAMA/LIBRA. These findings advance the theoretical foundation for dark matter interactions and provide a robust framework for future experimental design.

This paper extensively explores the concept of dark atoms, hypothetical stable lepton-like particles with a charge of $-2n$ (where n is any natural number) that form neutral bound states with n primordial helium nuclei. The discussion begins with the introduction of multiply charged stable particles. Next, the formation and evolution of dark atoms are examined, followed by a review of related constraints. The capture of dark atoms by the Earth and implications for direct dark matter search are subsequently discussed. Then, the quantum-mechanical description of bound states between dark atoms and ordinary nuclei is addressed. Moreover, procedures for systematic comparisons with this model, which have general interest, are presented considering the DAMA published results on the dark matter annual and diurnal modulation signatures as a benchmark.