SS symmetry

Article
Numerical Modeling of the Interaction of Dark Atoms with
Nuclei to Solve the Problem of Direct Dark Matter Search
Timur Bikbaev 1, *,† , Maxim Khlopov 1,2,3, *,† and Andrey Mayorov 1,†

1 Institute of Nuclear Physics and Engineering, National Research Nuclear University MEPhI,
115409 Moscow, Russia; agmayorov@mephi.ru
2 Institute of Physics, Southern Federal University, Stachki 194, 344090 Rostov-on-Don, Russia
3 Virtual Institute of Astroparticle Physics, 75018 Paris, France
* Correspondence: bikbaev.98@bk.ru (T.B.); khlopov@apc.univ-paris7.fr (M.K.)
† These authors contributed equally to this work.

Abstract: 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
Citation: Bikbaev, T.; Khlopov, M.; «dark atoms» and nuclei is developed through the continuous inclusion of realistic features of quantum
Mayorov, A. Numerical Modeling of
mechanics in the initial classical three-body problem involving the X-particle, the helium nucleus,
the Interaction of Dark Atoms with
and the target nucleus. This approach yields a comprehensive numerical model that encompasses
Nuclei to Solve the Problem of Direct
nuclear attraction and electromagnetic interaction between the «dark atom» and nuclei. Finally, this
Dark Matter Search. Symmetry 2023,
model aids in supporting the interpretation of the results obtained from direct underground dark
15, 2182. https://doi.org/10.3390/
sym15122182
matter experiments through the lens of the «dark atom» hypothesis.

Academic Editors: Vasilis K. Keywords: composite dark matter; stable charged particles; XHe; X-helium; «dark atoms»;
Oikonomou, Konstantin Zioutas and
low-energy bound state; nuclear interactions; Coulomb interaction
Kazuharu Bamba

Received: 17 October 2023

Revised: 1 December 2023
Accepted: 9 December 2023 1. Introduction
Published: 11 December 2023
According to hypotheses that claim dark matter consists of particles, these particles
are predicted beyond the framework of the Standard Model (SM) of elementary particle
physics. In recent decades, theories have been developed that extend the Standard Model
Copyright: © 2023 by the authors.
and propose various stable particles as candidates for dark matter—for example, supersym-
Licensee MDPI, Basel, Switzerland. metric (SUSY) particles and weakly interacting massive particles (WIMPs). These theories
This article is an open access article also have advantages in addressing the internal problems of the Standard Model [1–5].
distributed under the terms and However, the absence of experimental detection of such dark matter candidates
conditions of the Creative Commons stimulates research into a broader range of physics beyond the Standard Model. Particularly
Attribution (CC BY) license (https:// considering possible non-supersymmetric solutions aimed at reducing the divergence of
creativecommons.org/licenses/by/ the Higgs boson mass and explaining the physical nature of dark matter. Composite Higgs
4.0/). models, such as those based on the Walking Technicolor (WTC) hypothesis, offer such

Symmetry 2023, 15, 2182. https://doi.org/10.3390/sym15122182 https://www.mdpi.com/journal/symmetry

Symmetry 2023, 15, 2182 2 of 30

a solution. The application of the WTC hypothesis can also lead to a new approach to
understanding dark matter and revealing its composite nature [6–8]. In particular, this
approach assumes the existence of new stable, electrically charged particles. But it is
known that dark matter does not emit electromagnetic radiation, while charged particles
are sources of such radiation. Therefore, neutral weakly interacting elementary particles are
usually considered as candidates for dark matter. However, if stable charged particles form
composite neutral objects, they can also serve as dark matter of the universe. Studying such
neutral objects may provide the opportunity to determine the properties of the charged
stable particles from which they are composed.
Thus, Glashow proposed the existence of composite neutral dark matter, which is
formed in the early stages of the universe’s evolution as a tera-helium, consisting of new
charged particles. Glashow’s model is an extension of the Standard Model by introducing
an additional symmetry group, SU (2), which is associated with the presence of heavy
partners of ordinary particles (tera-partners) [9]. This extension allows for the resolution of
problems in the Standard Model, such as the violation of CP–parity in strong interactions
and the problem of neutrino mass. However, even before the establishment of the necessary
temperature of 25 keV, required for the formation of tera-helium and the annihilation and
binding of incomplete annihilation products, all the free tera-electrons E− , including those
that make up tera-helium, are captured by primary helium. As a result, positively charged
ions (4 HeE− )+ are formed, which prevents the implementation of this model [7].
In modern models of composite dark matter, the issues arising in Glashow’s model
can be resolved. For example, by introducing a new particle with a charge of −2, denoted
as O−− (in general, the charge could be −2n, where n—is a natural number, then we
will denote the particle as X), exceeding the number of its antiparticles, the problems
of Glashow’s model can be avoided. At a temperature of 100 keV, O−− forms a bound
state with primary helium, which is called the OHe or «dark atom» [6,10]. In models
where there are four or five generations of fermions, it is possible to form an excess of
antiparticles [11,12]. In such cases, there can exist a stable state with a charge of −2, similar
to tera-helium. Such a state can also bind with primary helium and form neutral OHe [13].
«Dark atoms» are considered as the promising candidates for the role of the composite
dark matter, which satisfy all the criteria necessary to reproduce the cosmological data on
which modern cosmology is based [14,15]. Namely, in spite of their nuclear interacting
nature, the gas of «dark atoms» decouples from plasma and radiation before the stage of
matter dominance and supports the development of gravitational instability. It provides
conditions for the growth of small initial density fluctuations, leading to the formation of the
large-scale structure of the Universe consistent with the data on the observed anisotropy
of the cosmic microwave background (CMB) radiation [14–17]. Specifics of the «dark
atom» nature of dark matter lead to a nontrivial «warmer than cold dark matter» scenario
of structure formation, which needs special studies but in any case is compatible with
the data of the precision cosmology [15]. The hypothesis of the existence of O-helium
atoms can explain the contradictory results in direct dark matter search experiments. They
are associated with the peculiarities of interaction between «dark atoms» and matter in
underground detectors [18]. For instance, the positive results from the DAMA/NaI and
DAMA/LIBRA experiments, indicating the detection of dark matter particles, contradict
the negative results from other experiments such as XENON100, LUX, and CDMS in
their direct search for dark matter particles. In preliminary qualitative calculations, it
was shown that «dark atoms» form low-energy bound states with nuclei of intermediate
masses, excluding such a binding with nuclei of heavy elements [10]. In addition, taking
into account the scalar and isoscalar nature of XHe, binding to such a low-energy «dark
atom» nucleus state can go only due to an electric dipole E1 transition with a violation of
isotopic invariance. This transition is proportional to the square of the relative velocity,
and hence the temperature, and is therefore suppressed in cryogenic experiments [10]. All
this explains the lack of positive results in other experiments, since their strategy is aimed
Symmetry 2023, 15, 2182 3 of 30

at searching for the effects of the recoil nuclei, which may interpret effects of energy release
as background events.
In the aforementioned Walking Technicolor (WTC) hypothesis, which is a composite
Higgs model, there are mechanisms for particle mass generation and spontaneous sym-
metry breaking of the electroweak interaction. This model suggests the existence of a
new type of interaction that binds a new type of quark [19,20]. Owing to the electroweak
SU (2) charge of the techni-particles WTC model provides the generation of an excess of
−2n charged stable techni-particles over their +2n charged antiparticles. This excess is
provided by sphaleron transitions, which establish its balance with baryon asymmetry.
At the same time, the electric charge of the Universe is conserved, since the charge of the
excessive −2n charged particles is compensated by the charge of other stable charged parti-
cles, namely, by the excess of protons [21]. This correspondence between the excess of stable
particles with a charge of −2n and baryon asymmetry can explain the observed ratio of
baryonic and dark matter densities [6–8]. Moreover, at the masses of these techno-particles
in the 1 TeV range, their contribution to the total density corresponds to the observed dark
matter. Since the mass of these heavy −2n charged particles determines the mass of «dark
atoms», therefore, they explain the observed density of all dark matter.
In WTC, the charge of «new» physics particles is not fixed. However, fractional charges
are limited as free quarks are not observed. A major problem for scenarios involving
hypothetical stable electrically charged particles is their absence in the surrounding matter.
If such particles exist, they must be bound to ordinary matter and form anomalous isotopes
with unusual Z/A ratios (number of protons to mass number). The main challenge for
these scenarios is the suppression of an abundance of positively charged particles bound to
electrons. They could behave like anomalous isotopes of hydrogen or helium. There are
significant experimental constraints on such isotopes, particularly anomalous hydrogen,
which strongly restrict the possibility of stable positively charged particles existing [22].
Thus, positively charged particles cannot be candidates for dark matter particles. The same
problem arises if the model predicts the existence of stable particles with negative, odd
charges. These particles combine with primordial helium to form ion-like systems with a
charge of +1. Then, they undergo recombination with electrons to form atoms of anomalous
hydrogen [23]. Therefore, stable negatively charged particles can only have a charge of
−2n [24].
In this study, we consider a scenario where dark matter is composed of hypothetical
stable particles, X, which evade detection in experiments, because they form neutral atom-
like states called X-helium with primordial helium [25]. Since all these models also predict
corresponding antiparticles with a charge of +2n, the cosmological scenario must provide
a mechanism for their suppression. This may occur through a charge asymmetry associated
with an excess of −2n charged particles [1]. However, the overall electric neutrality of
the Universe is maintained because the surplus charge is balanced out by an equivalent
surplus of baryons with positive charge. As a result, the antiparticles with positive charge
can undergo efficient annihilation in the early stages of the Universe. There are several
models that predict such stable particles with a charge of −2n [12,19,26].

2. X-helium Atoms
A «dark atom» refers to a system composed of −2n charged particle (for n = 1, this
corresponds to O−− ) and n 4 He nuclei, which are bound together by the Coulomb force.
The specific structure of this bound state is determined by the parameter
a ≈ Zα ZX αAm p RnHe , where α represents the fine structure constant, ZX and Zα denote the
charge numbers of particle X and n nuclei of He, respectively, m p represents the mass of
a proton, A corresponds to the mass number of the n-nucleus of He, and RnHe represents
the radius of the corresponding nucleus. The physical meaning of the parameter a (not α)
is related to the ratio of the Bohr radius of a «dark atom» to the radius of the n–α-particle
nucleus. For example, for OHe, this parameter shows the ratio of the Bohr orbit for the
α-particle in a «dark atom» to the radius of the helium nucleus. If the Bohr orbit of XHe
Symmetry 2023, 15, 2182 4 of 30

turns out to be smaller than the size of the n-nucleus of helium, then the «dark atom» has
the structure of Thomson’s atom.
When the value of a falls within the range of 0 < a < 2, the bound state exhibits a Bohr
atom-like structure with a negatively charged particle at the core and a nucleus orbiting it
in a manner similar to Bohr’s model. On the other hand, when 2 < a < ∞, the bound states
resemble Thomson’s atoms, where the nucleus vibrates around a heavy massive negatively
charged particle.
If Zα = 2 and ZX = −2, the model treats the α-particle as a point-like particle and
travels along the Bohr radius. Consequently, the binding energy for OHe, involving the
point-like charge of 4 He, can be calculated using the expression:
2 Z 2 α2 m
ZX α He
I0 = ≈ 1.6 MeV, (1)
2
where m He represents the mass of the α-particle.
In «dark atoms» OHe, the Bohr radius of rotation for He is equivalent to [10]:

h̄c
Rb = ≈ 2 · 10−13 cm. (2)
ZX Zα m He α

According to the X-helium model, X can either exhibit lepton-like behavior or act as a
distinct cluster of heavy quarks belonging to new families that have suppressed interaction
with hadrons [13]. Furthermore, experimental evidence suggests that the minimum mass
of multiply charged stable particles is approximately 1 TeV [27]. In this work, the mass of
the X particle is always taken to be equal to 1 TeV.
The XHe hypothesis is notable for its simplicity, as it relies on only one parameter of
«new» physics—the mass of the X particle. However, it necessitates a thorough understand-
ing of known nuclear and atomic physics, which have not yet been applied to non-classical
bound systems like «dark atoms» XHe [28]. Research focused on the active influence of
this type of dark matter on nuclear transformations is crucial and leads to advancements in
the nuclear physics of X-helium. This research is particularly important when assessing the
quantitative role of «dark atoms» in primordial cosmological nucleosynthesis and stellar
evolution [10].
The deceleration of cosmic XHe as it passes through terrestrial matter makes it chal-
lenging to directly detect dark matter particles using methods that rely on recoiling effects
from WIMP–nucleus collisions. The parameters of X-helium atoms, both Thomson and
Bohr, ensure effective slowing down of the flow of cosmic «dark atoms» to thermal energies
and slow diffusion to the center of the Earth. Therefore, the search for dark matter particles
using the effects of the recoil nucleus ([29]) is impossible in this case. And the interpre-
tation of the results of the DAMA experiment requires another explanation, which we
propose and try to justify. When slow X-helium atoms interact with nuclei, they can form
low-energy bindings between each other. Within the uncertainty range of nuclear physics
parameters, there is a specific range where the binding energy in the OHe-Na system falls
within the 2–4 keV interval [1,30]. When «dark atoms» are captured and enter this bound
state, they release energy. This energy can be observed as an ionization signal in detectors
like DAMA. The concentration of XHe in underground detectors is determined by the
balance between the cosmic influx of dark matter atoms and their diffusion toward the
center of the Earth. The presence of X-helium in the Earth’s crust is quickly regulated
through the dynamics of interactions between «dark atoms» and matter, considering the
incoming cosmic XHe and changes in its flux. As a result, the capture rate of «dark atoms»
should exhibit annual modulations, which would be reflected in the annual modulation
of the ionization signal produced by these reactions. A significant consequence of this
proposed explanation is the emergence of anomalous superheavy sodium isotopes in the
detector material of DAMA/NaI or DAMA/LIBRA. The mass of these isotopes is approx-
imately an order of magnitude higher than that of regular isotopes of sodium. However,
Symmetry 2023, 15, 2182 5 of 30

the occurrence of anomalous superheavy iodine isotopes is unlikely, as calculations indicate

that it is not favorable for X-helium atoms to form low-energy bound states with these nu-
clei [10]. If the atoms of these unusual isotopes are not fully ionized, their ability to move is
determined by atomic cross-sections and is approximately nine orders of magnitude lower
than that for OHe [10]. This means that they will remain in the detector. Thus, analyzing
the material using mass spectroscopy can provide further confirmation of the presence of
an O-helium component in the DAMA signal. The methods used for this analysis should
consider the delicate nature of the bound states of OHe-Na, as their binding energy is only
a few keV [28,31].
When dark matter atoms collide with matter, particularly in regions with higher
concentrations of XHe in the central part of the Galaxy, it can cause the excitation of X-
helium. As a result, the excess of the positron annihilation line observed by I NTEGRAL
in the central part of the galaxy can be attributed to the creation of pairs emitted by the
excited «dark atoms» XHe that are formed through these collisions [28,32].
The primary issue with XHe atoms is their potential strong interaction with matter.
This arises from the unshielded nuclear attraction of helium to the nuclei of matter. This
interaction has the potential to disrupt the bound system of dark matter atoms and result
in the creation of abnormal isotopes. Strict experimental limitations on the concentration of
these isotopes exist in terrestrial and marine environments [22]. To prevent an excessive
production of abnormal isotopes, it is hypothesized that the effective interaction between
XHe and matter nuclei should have a shallow well and a barrier that hinders the fusion of
He and/or X with the nucleus (see Figure 1). Facing the challenge of being a three-body
problem, an exact analytical solution is absent. In order to evaluate the physical meaning of
the proposed scenario with a potential barrier and a shallow well in the effective interaction
potential, in this work, we developed our semiclassical approach, which is determined
by the fact that the nontrivial nature of the problem requires, first of all, clarification of
the possibility of implementing this scenario. Unlike ordinary atoms, in a dark atom, we
have a leptonic core and a nuclear interacting shell. In this case, the usual approximations
of atomic physics do not work. Therefore, we wanted to consistently approach a correct
quantum mechanical description starting with a semiclassical model, which allows us
to figure out the most essential points of the proposed dark atom scenario. Hence, this
study puts forward a numerical approach to characterize the interaction between XHe
and matter nuclei, using a numerical model to reconstruct the form of the corresponding
effective potential.

Figure 1. The interaction between XHe and the nucleus of matter is characterized by the effective
potential [10].
Symmetry 2023, 15, 2182 6 of 30

3. Bohr’s Model
3.1. Simulation of O-Helium
The OHe system consists of two particles, namely the He nucleus and the O−− particle.
These particles are bound together and considered point-like. In this system, a spherical
coordinate system is centered at the O−− particle. The He nucleus moves randomly along
the surface of a sphere with a radius equal to the atom’s radius Rb . The He nucleus
maintains a constant velocity known as the Bohr velocity Vα .
The speed of the α-particle moving along the Bohr orbit is equal:

h̄c2 cm
Vα = ≈ 3 · 104 . (3)
m He Rb s

Let us consider a numerical simulation scheme for the dynamic system of OHe:
(1) Within the OHe bound system, the α-particle is characterized by two independent
degrees of freedom, specifically the polar angle, θ, and the azimuthal angle, φ. The initial
values of these angles, θ0 and φ0 , are used to compute the initial components of the
α-particle’s position vector, which is denoted as r0 .
(2) The variations in the polar angle, dθ, and the azimuthal angle, dφ, are calculated as
the differences in angles while moving from the point ri−1 to the point ri on the surface of a
sphere within one iteration of time dt; here, the subscript i represents the iteration number:

Vα dt
  
dθ = 2n − 1 , (4)
Rb
s 2  2
Vα dt
− dθ 
Rb

dφ =
2n − 1 , (5)
cos θ
where n is a random variable with a uniform distribution on the interval from 0 to 1.
(3) We verify the condition for the change in angles:
2 2 
Vα dt 2
  
dθ + cos θdφ ≤ . (6)
Rb

This condition is essential to ensure that the calculated trajectory of the alpha particle,
based on the incremental angles dθ and dφ, does not surpass the actual distance covered by
the alpha particle on the sphere during time dt.
The positional components of the α-particle’s vector, r, are computed for each time
step, enabling the construction of its trajectory on the surface of sphere of the Bohr radius
(see Figure 2). This sphere, with a radius of Rb , illustrates the α-particle’s location through
blue dots, representing its positions between dt time intervals. The density of these points
on the sphere is determined by the number of iterations in the cycle.

3.2. OHe-Nucleus Coordinate System

Let us examine the OHe-nucleus system, which is composed of three charged, point-
like particles. The coordinate system is positioned at the center of the target nucleus A.
In the chosen reference system, O-helium moves in relation to the origin of coordinates.
Herewith, the α-particle moves along a sphere with Bohr radius Rb ; in the center of this
sphere, there is a particle O−− . We introduce the position vector O−− , ~r, and the posi-
tion vector of the α-particle, ~rα (see Figure 3). In this scenario, ~rα is determined in the
following manner:
~rα = ~r + ~Rb . (7)
Symmetry 2023, 15, 2182 7 of 30

Figure 2. The spatial distribution of the α−particle’s coordinates within the orbit that corresponds to
the ground state of the OHe system. Original authors’ figure taken from [33].

Figure 3. OHe−nucleus coordinate system. Original authors’ figure taken from [34].

Our task is to consider the interaction of OHe with the nucleus by constructing a set
of forces acting between all particles in the chosen coordinate system. We must take into
account the electromagnetic forces acting between O−− and the nucleus, O−− and He,
and He and the nucleus as well as the nuclear interaction between helium and the target
nucleus. This problem is formulated as a three-body problem without an exact analytical
solution. Thus, we propose a numerical approach to describe the listed interactions.

3.3. Simulation of the OHe-Nucleus System

The formulas for the Coulomb interaction between the α-particle and the target nucleus
and between O−− and the target nucleus are as follows:

ZZα e2~riα
~Fi eα = ~Fi eα (~riα ) = , (8)
3
riα

ZZo e2~ri
~Fi eZO = ~Fi eZO (~ri ) = , (9)
ri3
where Z, Zo are the charge numbers of the target nucleus and the O−− particle, respectively,
and index i indicates the iteration number.
Symmetry 2023, 15, 2182 8 of 30

The nuclear interaction between the He nucleus and the target nucleus is determined
N
by the nuclear force of the Woods–Saxon type, ~Fi α :

U0 riα − R Z ~riα
 
exp
~Fi αN p p riα
=−  !2
, (10)
riα − R Z

1 + exp
p

where R Z is the radius of the target nucleus, U0 is the depth of the potential well, and
p ≈ 0.55 fm is a constant parameter.
The total force acting on the «dark atom» of OHe, ~FiSum , is calculated as follows:

~FiSum = ~Fi eZO + ~Fiα , (11)

where ~Fiα is defined as the total force acting on the α-particle:

~Fiα = ~Fi eα + ~Fi αN . (12)

To calculate the total force acting on the «dark atom» OHe, we developed a numerical
method based on the distance between objects. This method will utilize the described
earlier model for describing the OHe system and apply it to the OHe-nucleus system
for calculations.
(1) To begin, we will establish the initial conditions for the calculation. These conditions
include the initial coordinates of O−− , denoted as [ x0 , y0 , z0 ] or simply r0 , as well as the
initial components of its velocity, denoted as [Vx0 , Vy0 , Vz0 ] or V0 , (i = 0).
(2) Next, we determine the i-th value of the increment of the α-particle momentum at
time interval dt, which is denoted as d~ Pαi :

d~Piα = ~Fiα dt. (13)

(3) The program checks for a condition to interrupt when the excess kinetic energy dTi
transferred to the α-particle exceeds the ionization potential of O-helium I0 . This condition
leads to the destruction of the bound OHe system:
2
d~Pi α
dTi = < I0 ≈ 1.6 MeV. (14)
2mα

(4) The new position of the particle O−− is calculated:

ri+1 = ri + Vi dt (15)

(5) Using the model for describing the system of the «dark atom» OHe and the new
position of the O−− particle, the new position of the α-particle, rαi+1 , is calculated.
(6) During every iteration, the program computes the total force exerted on the OHe
system, which is denoted as ~Fisum .
(7) The program calculates the change in momentum d~Pi of the OHe system, which is
equal to the change in momentum of the particle O−− overall:

d~Pi = ~Fisum dt. (16)

Symmetry 2023, 15, 2182 9 of 30

(8) The program calculates the change in velocity dV ~ i of the particle O−− using the
~
momentum increment d Pi , and this velocity change is used for subsequent finding of the
new speed, Vi+1 by adding it to the old value of the O−− particle velocity:

d~Pi
~i =
dV . (17)
mO−− + mα

Using the acquired data, the program restores movement trajectories of both the α-
particle and O−− particle in the XY plane (see Figure 4). The target nucleus, specifically
the Na nucleus, is depicted as a black circle in Figure 4, which displays the outcome of
the program. The α-particle trajectory is represented by blue dots, while the O−− particle
trajectory is shown as a red dashed line.

Figure 4. The movement trajectories followed by an α−particle and the O−− in the XY plane. Original
authors’ figure taken from [33].

Figure 4 displays one of the outcomes of our simulation. The figure illustrates how
the trajectory of O−− deviates from its initial path due to the Coulomb interaction between
the He nucleus and the target nucleus. This deviation occurs because when rotating in
Bohr’s orbit, there are times when the He nucleus approaches the origin and experiences
a stronger repulsion from the target nucleus compared to the attraction it receives from
the O−− particle. Additionally, the figure shows that the trajectory of O−− undergoes
oscillations. These oscillations are caused by an additional nuclear interaction between the
α-particle and the nucleus, which leads to an attraction of the α-particle. As a result, as the
α-particle approaches the nucleus, this force becomes stronger, causing further distortion in
the trajectory of O−− .

3.4. Stark Effect in the OHe-Nucleus System

In our numerical Bohr model, we restrict the rotation orbit of He within the OHe atom,
preventing its polarization. And we can observe the Coulomb repulsion between the «dark
atom» and the target nucleus. Conversely, when an alternating external electric field is
present, such as the one created by the target nucleus, we expect to see the Stark effect,
which results in the polarization of XHe. We introduce the interaction dipole moment δ
caused by the Stark effect to incorporate this effect into our semiclassical numerical model.
By manually including δ in our numerical model, we can calculate the Stark force, which
can be easily derived from the potential. This potential is determined by the same dipole
moment. The appearance of δ is a result of the nuclear force and Coulomb force acting on
Symmetry 2023, 15, 2182 10 of 30

the nHe nucleus from the outer nucleus. These forces are balanced by the Coulomb force
between the particles of the dark matter atom. From this, we can derive an expression for δ:

Zα ~E ~FαN
~δ = + , (18)
ZX 4/3πρ eZX 4/3πρ

Zα e
here, ~E represents the strength of the external electric field, and ρ = denotes the
4/3πR3nHe
charge density of the nHe nucleus.
In an external electric field, a moment of force acts on an electric dipole, which tends
to rotate it so that the dipole moment turns along the direction of the field. The potential
energy of a polarized XHe atom, the Stark potential, in an external electric field is equal to:

USt = eZα (~E · ~δ). (19)

In turn, the Stark force is calculated using the Stark potential as follows:

~FSt = − grad USt . (20)

After adding the Stark force to the numerical model, using the obtained data, it is also
possible to reconstruct the trajectories of the α-particle and the O−− particle for different
impact parameters. For example, for zero and non-zero impact parameters, Figures 5 and 6
demonstrate the result of the program. The black circle shows the location of the target
nucleus, for which we hereinafter take the sodium nucleus, since in the DAMA/NaI or
DAMA/LIBRA experiment, the detector substance consists of NaI, and the results of our
simulation may be useful for interpreting the results of this experiment. The blue dots
show the trajectory of the α-particle, and the red dashed line shows the trajectory of the
O−− particle in the XY plane.

Figure 5. The movement trajectories followed by an α−particle and the O−− particle in the XY plane
for zero impact parameter.
Symmetry 2023, 15, 2182 11 of 30

Figure 6. The movement trajectories followed by an α−particle and the O−− particle in the XY plane
for non-zero impact parameter.

1. Impact parameter β = 0 fm (see Figure 5).

The sum of the radii of the helium nucleus and the target nucleus Na, R AHe = 5.2 fm.
Examining the minimum distance between the α-particle and the target nucleus in Figure 5,
Rαmin = 5.5 fm, that is, the α-particle does not hit the target nucleus, and elastic scatter-
ing occurs.
The maximum value of the dipole moment is δmax = 1.71 · 10−12 cm.
The average value of the dipole moment is δmean = 8.97 · 10−18 cm.
Figure 7 plots the dependence of the dipole moment on the distance between the O−−
particle and the target nucleus Na at zero impact parameter. It can be seen that the closer
the dark matter atom is to the target nucleus, the greater the polarization it experiences,
which is consistent with the theory.

Figure 7. Dependence of the dipole moment on the distance between the O−− particle and the target
nucleus Na (dot line) in the Bohr model at zero impact parameter.

Figure 8 plots the total interaction potential between OHe and the target nucleus Na
depending on the O−− particle radius vector at zero impact parameter. It is clear from the
Symmetry 2023, 15, 2182 12 of 30

figure that it qualitatively coincides with the theoretically expected form of the effective
interaction potential between a dark matter atom and the nucleus of matter.

Figure 8. The total interaction potential between OHe and the target nucleus Na (dot line) depending
on the radius vector of the particle O−− at zero impact parameter.

2. Impact parameter β = 1 fm (see Figure 6).

The minimum distance between the α-particle and the target nucleus with a non-zero
impact parameter Rαmin = 6.3 fm; that is, elastic scattering also occurs in this case.
The maximum value of the dipole moment is δmax = 7.31 · 10−13 cm.
The average value of the dipole moment is δmean = 3.37 · 10−15 cm.
Figure 9 plots the dependence of the dipole moment on the distance between the
particle O−− and the target nucleus Na for a non-zero impact parameter. Again, we can
see that the closer a «dark atom» is to the target nucleus, the more polarized it becomes.

Figure 9. Dependence of the dipole moment on the distance between the particle O−− and the target
nucleus Na (dot line) in the Bohr model for a non-zero impact parameter.

Figure 10 plots the total interaction potential between OHe and the target nucleus Na
depending on the particle radius vector O−− for a non-zero impact parameter. This total
interaction potential also qualitatively coincides with the theoretically expected form of the
effective interaction potential between «dark atom» and the nucleus of matter.
Symmetry 2023, 15, 2182 13 of 30

Figure 10. The total interaction potential between OHe and the target nucleus Na (dot line) depending
on the radius vector of the particle O−− for a non-zero impact parameter.

However, there are drawbacks to the Bohr atom model. For instance, our numerical
model does not explicitly account for the Coulomb force between helium and O−− . Addi-
tionally, the rotation orbit of helium in the OHe atom is manually fixed, which prevents
its natural polarization due to the Stark effect. This necessitates the introduction of an
analytical formula to calculate the dipole moment and the Stark force, leading to a decrease
in the resulting accuracy. On the other hand, these issues can be resolved when considering
the Thomson model of the atom. In this approach, helium is not treated as a point charge,
moving randomly along a fixed Bohr orbit, but rather as a charged ball within which the
O−− particle can oscillate. Furthermore, it is important to note that the case of −2 charged
particles is just one specific scenario. The particle in the “new” physics we are examining
can have a charge of −2n, forming X-helium «dark atoms» with n nuclei 4 He. These
X-helium «dark atom» themselves, starting from n = 2, are Thomson atoms.

4. Thomson’s Model
4.1. Simulation of X-helium
The X-helium «dark atom» comprises two interconnected components: the n-helium
nucleus and the X particle. We establish a spherical coordinate system at the center of
the charged n-helium nucleus, representing a charged sphere, within which resides the
point particle X. When subjected to external disturbances, such as external forces causing a
non-zero distance between the n-helium and X, the X particle initiates oscillations within
the nHe nucleus (in reality, the nHe is significantly lighter than X, resulting in the nuclear
drop oscillating around X).
The Coulomb interaction potential and corresponding force between the n-helium and
X are expressed by the following equations:

4e2 n2

−

 for R XHe > R He ,
 R XHe
UXHe ( R XHe ) = 2 2 2
! (21)
− 4e n 3 − R XHe

for R XHe < R He ,
R2He

 2R
He

4e2 n2

− 3 ~R XHe


 for R XHe > R He ,
~FXHe ( R XHe ) = R XHe
2 n2 (22)
4e
− 3 ~R XHe


 for R XHe < R He ,
R He
Symmetry 2023, 15, 2182 14 of 30

where R XHe is the distance between the X particle and the nHe nucleus, and R He is the
radius of the n-helium nucleus.
Let us consider a numerical simulation scheme for the dynamic XHe system.
(1) We will start with the following initial conditions: the particle X has an initial
coordinate R ~ 0X = 0, and its initial velocity is set to the thermal speed in the medium,
!1/2
3kT
V0X = . Here, Mnuc represents the mass of the target nucleus (e.g., Na), T is the
Mnuc
temperature (assumed to be 25 degrees Celsius), and k is Boltzmann’s constant. By using
these initial conditions, we can calculate the initial force acting on the particle X.
(2) Next, we determine the increment of the components of the particle X’s radius
vector, denoted as dri , over a time interval dt:

dri = ViX dt. (23)

(3) Using the calculated increment from step 2, we can determine the i + 1 value of the
components of the radius vector for particle X, which is denoted as ri+1 :

ri+1 = ri + dri . (24)

(4) The program determines the force applied to the X particle, which is denoted as
~Fi XHe . This force is then used to determine the momentum increment d~Pi for particle X:

d~Pi = ~Fi XHe dt. (25)

(5) Using the momentum increment d~Pi , we can calculate the velocity increment dV~iX
for the X particle. This increment is added to the velocity from the previous iteration to
find the new velocity for use in the next iteration:

d~Pi
~i =
dV . (26)
X
mX

By analyzing the data, we can create a graph showing how the magnitude of particle
X’s radius vector changes over time. This graph, shown in Figure 11, demonstrates that
particle X undergoes oscillations within the nucleus nHe with a period of approximately
2 · 10−20 s. These oscillations occur because the Coulomb force between the nucleus nHe
and particle X acts to bring particle X back toward the center of the nucleus and counteract
any external disturbances. The fact that R XHe < 1 fm indicates that the X-helium system
is stable.

Figure 11. The relationship between the magnitude of the particle X’s radius vector and time t (dot
line), for n = 1. Original authors’ figure taken from [35].
Symmetry 2023, 15, 2182 15 of 30

4.2. Simulation of the XHe–Nucleus System

Simulation of the interaction of the XHe «dark atom» with the target nucleus occurs
in the XHe–nucleus coordinate system, which is similar to the OHe–nucleus coordinate
system described in Section 3.2 of this study. However, these systems differ in that the
distance between the «dark atom» particles X and nHe is no longer fixed and is not
equivalent to the Bohr radius. Therefore, the positions of the nHe particle, r He , and the X
particle, r, are determined independently. In this case, the distance between the particle X
and nHe, r XHe , is calculated by the formula:

~r XHe = ~rα −~r (27)

The types of interactions acting in the XHe -nucleus system between particles are
identical to the forces described in Section 3.3 of this study. However, two additional forces
arise, which are equal in magnitude but opposite in direction. These forces represent the
Coulomb force between X and nHe (refer to Formula (22)). The force acting on nHe is
XHe XHe XHe
denoted as ~Fi α , while the force acting on X is denoted as ~Fi X = −~Fi α .
X
The total force to which particle X is subject, ~Fi Sum , is defined as follows:
X e XHe
~Fi Sum = ~Fi ZO + ~Fi X . (28)

The total force acting on nHe, ~Fiα , is determined by the formula:

~Fiα = ~Fi eα + ~Fi αN + ~Fi αXHe . (29)

To calculate the total force acting on the XHe «dark atom», we are developed a
numerical method.
(1) The initial conditions we use include the initial coordinates X and nHe, represented
by ~
r0 = ~r0α , as well as their initial velocities. We set the initial velocities equal to the thermal
!1/2
3kT
speed in the medium, which is given by VX0 = Vα0 = .
Mnuc
(2) The i-th value of the momentum increment nHe particle, d~Piα , and X particle, d~Pi ,
taken over the time interval dt is determined:

d~Piα = ~Fiα dt, (30)

X
d~Pi = ~Fi Sum dt. (31)
(3) By using the increment d~Piα and d~Pi , we can determine the velocities of the nucleus
nHe and X at the i + 1 time step. These velocities are denoted as V~ α and V ~X :
i +1 i +1

d~Piα
~α = V
V ~α + , (32)
i +1 i
m He

d~Pi
~X = V
V ~X + . (33)
i +1 i
mX
(4) Next, the i + 1 value of the radius vector for X and nHe can be calculated:

~ α dt,
~ri+1 = ~ri + V (34)
i +1

~ X dt,
~rαi+1 = ~rαi + V (35)
i +1

(5) Then, the program calculates the total force acting on the X particle, represented
X
by Fi Sum , as well as the total force acting on nHe, denoted as ~Fiα .
~
Symmetry 2023, 15, 2182 16 of 30

Based on the obtained data, we can plot the relationship between the radius vector of
the particle X and the radius vector of the n-helium nucleus (see Figure 12). Additionally,
we can plot the dependence of the total interaction potential between nHe and the target
nucleus Na on rα for n = 1 (see Figure 13).

Figure 12. Dependence of r on rα (blue line) in the Thomson model for n = 1. Original authors’ figure
taken from [35].

Figure 13. Dependence of the total interaction potential of nHe, at n = 1, with the target nucleus Na
on rα (pink dot line). Original authors’ figure taken from [35].

From Figures 12 and 13, it is clear that the XHe system moves toward the target
nucleus as a bound system. And the X particle is slightly ahead of the nHe nucleus, since
the radius vector of the X particle at each point is less than the radius vector of the nucleus
nHe (see Figure 12). The nHe nucleus flies behind oscillating; i.e., polarization of the «dark
atom» is observed. It can be seen that at a sufficiently close distance from the target nucleus,
the nuclear force between the n-helium nucleus and the target nucleus becomes quite strong.
Therefore, it exceeds the Coulomb repulsion of nHe by the target nucleus and n-helium,
rushing ahead of the X particle and penetrating the target nucleus (see Figure 13).
Following this, in our numerical model, we included the Coulomb force between
the nHe and the target nucleus, as well as the Coulomb force between the X particle and
the target nucleus, using a similar approach as in Formula (22). Namely, we introduced
a condition that causes a change in the form of force when either the nHe or X particles
penetrate the target nucleus.
Symmetry 2023, 15, 2182 17 of 30

From analyzing the trajectories obtained, we can identify two distinct scenarios for
n = 1. When the impact parameter is zero, the XHe atom passes through the target nucleus
Na; then, it reverses its direction and flies back in the opposite direction (see Figure 14).

Figure 14. Dependence of the total interaction potential of nHe, for n = 1, with the target nucleus Na
on rα (pink line) at zero impact parameter. Original authors’ figure taken from [35].

According to the theory, the interaction between slow X-helium atoms and nuclei can
result in the formation of a low-energy bound state. When «dark atoms» are captured in
this bound state, energy is released, which can be detected as an ionization signal in the
DAMA detector. Therefore, the low-energy bound state of XHe and the nucleus can be
considered as a three-body oscillating system.
In Figures 15 and 16, we can observe that when the XHe atom with a non-zero impact
parameter β = 0.2 · 10−13 cm collides with the target nucleus, a three-body oscillatory
system is formed. The minimum distance between the n helium atom and the target
nucleus Na in this case is Rαmin = 6.8 · 10−16 cm.
In Figure 15, the black circle represents the target nucleus Na, while the yellow
star and green diamond represent the initial positions of nHe and the X particle, respec-
tively. The blue dots and red dotted line show the trajectories of nHe and the X particle,
respectively.
The maximum value of the dipole moment is δmax = 1.43 · 10−12 cm.
The average value of the dipole moment is δmean = 5.39 · 10−13 cm.
During trajectory analysis, it was discovered that changes in the mass of X do not
have any effect on the outcome. The outcome is dependent on the impact parameter and
the initial velocity of the system. However, regardless of their values, the cloud of particle
coordinates always ends up inside the target nucleus. This is likely reasonable because a
zero force balance for helium can only be achieved in the region where the nuclear and
Coulomb forces are balanced. Outside of this region, nuclear force is minimal. As a result,
Coulomb polarization occurs in XHe up to the boundary of the nucleus, while nuclear
polarization only occurs inside.
When modeling using the Thomson atom approximation, several effects were ob-
served. When the impact parameter is zero, the XHe atom passes through the target
nucleus; then, it returns back and flies in the opposite direction. With a non-zero impact
parameter, the XHe atom collides with the target nucleus, forming a vibrational system
of three bodies. This is expected during the formation of a low-energy bound state of
XHe with the nucleus. Additionally, in the Thomson model, the polarization of the «dark
atom» automatically occurs due to the Stark effect, whereas in the Bohr model, it had to be
manually introduced.
Symmetry 2023, 15, 2182 18 of 30

Figure 15. Trajectories of motion of a n-helium nucleus and X particle, for n = 1, in the XZ plane
with a non-zero impact parameter in the Thomson model. Original authors’ figure taken from [35].

Figure 16. Dependence of the total interaction potential of nHe, for n = 1, with the target nucleus Na
on rα (pink dot line) for a non-zero impact parameter. Original authors’ figure taken from [35].

However, the disadvantage of the Thomson approximation is that particle oscillations

occur inside the target nucleus. And elastic scattering is not observed, the presence of
which is very important for the existence of the XHe hypothesis. Herewith, in Bohr’s
atom approach, the opposite was true. In the Bohr model, the quantum mechanical
connection between the particles of the «dark atom» is preserved. This quantum mechanical
connection is expressed in the form of a fixed Bohr orbit of the α-particle, but this did not
allow obtaining automatic polarization of the «dark atom». In Thomson’s model, such a
connection is lost. This happens because in this model, XHe represents two independent
particles, between which the Coulomb force clearly acts, which can be introduced due to
the automatically appearing polarization. That is, you can not introduce polarization «by
hand», but in this case, the quantum-mechanical connection in XHe is lost. Or you can
preserve this connection, but then you have to calculate the length of the dipole moment in
an unnatural way, which affects the accuracy of the results. Therefore, in Thomson’s model,
it is necessary to introduce a connection between helium and the X particle. In addition,
Symmetry 2023, 15, 2182 19 of 30

in order to more accurately describe the interaction of XHe with the nucleus of matter,
which can lead to solving the problem of X-helium entering the nucleus and leading to the
dominance of elastic interactions, it is also necessary to manually introduce the Stark force,
as in the Bohr model, but already using the naturally obtained dipole moment length.

4.3. Stark Force in Thomson’s Model

Since in the Thomson model, the radius vector of the nHe nucleus, r He , and the
X particle, r, are determined independently, the distance between X and nHe, r XHe , is
calculated as the difference between the radius vectors of these particles (see Formula
(27)). But at the same time, r XHe also determines the polarization of the «dark atom» in the
Thomson model; therefore, the value of the dipole moment is equal to:

δ = |~r XHe |. (36)

In order to add the Stark force to the Thomson model, in each i-th iteration, the
iteration step for the nHe nucleus is determined, hi = |~rαi+1 | − |~rαi |, and using the calculated
Coulomb force between nHe and the target nucleus of the substance, ~Fie+1α , the potential
and Stark force are calculated:

USti+1 = Fie+1α |~r XHei+1 |, (37)

USti+1 − USti
Fi+1St = − . (38)
hi
Consequently, the total force acting on nHe now consists of the nuclear force, the
Coulomb force between X and nHe, the Coulomb force between nHe and the nucleus of
the substance, and the Stark force.
To strengthen the connection between dark matter atom particles, we also changed
the calculation of the i + 1 values of the velocities of the nHe nucleus and the X particle,
~ α and V
V ~ X , by adding the following condition: if |~r XHe | < R He , i.e., if the X particle is
i +1 i +1
inside nHe, where R He is the radius of the n-helium nucleus, then the calculation is carried
out using the following formulas:

d~Piα
~α = V
V ~α + , (39)
i +1 i
m X + m He

d~Pi
~X = V
V ~X + . (40)
i +1 i
m X + m He
Thus, the momentum increments of nHe, d~Piα , and particles X, d~Pi , are divided by
the entire mass of XHe. If |~r XHe | > R He , i.e., if particle X is no longer inside nHe, then
the increment in momentum is divided by the masses of the corresponding particles (see
Formulas (32) and (33)).
From the analysis of trajectories in the Thomson model with the Stark force, for n = 1,
i.e., when the X particle is an O−− particle, and the nHe nucleus is an α-particle, two
characteristic cases can be distinguished independent of the impact parameter: elastic
interaction and inelastic interaction, when the X particle enters the nucleus.
Elastic interaction with a non-zero impact parameter can be seen in Figure 17, which
shows the trajectories of motion of nHe and X particles in the XZ plane during interaction
with the Na nucleus.
In Figure 17, the blue line and the red dotted line show the trajectories of n- helium
and the X particle during elastic interaction in the XZ plane, respectively. The green
diamond shows the initial position of the «dark atom» particles. The figure shows that
as a result of elastic scattering, XHe particles change their direction of motion to the
opposite. The X-helium atom experiences greater polarization the closer it approaches the
target nucleus.
Symmetry 2023, 15, 2182 20 of 30

Figure 17. Trajectories of the nHe (blue line) and X particles (red dot line), for n = 1, in the XZ plane
with a non-zero impact parameter, for elastic interaction.

The impact parameter and the minimum distance between the α-particle and the target
nucleus, corresponding to the interaction shown in Figure 17, are equal to
β = 0.5 · 10−13 cm , Rαmin = 1.3 · 10−12 cm.
Figure 18 plots the dependence of the magnitude of the dipole moment modulus on
the distance between the particle X and the target nucleus Na for elastic interaction and a
non-zero impact parameter.
The maximum value of the dipole moment for the case in Figure 17 is
δmax = 9.08 · 10−14 cm, while the average value of the dipole moment is δmean = 2.33 ·
10−15 cm (see Figure 18).
It is also possible to plot the dependence of various potentials corresponding to the
forces acting between particles in the XHe–nucleus system, depending on the distance
between nHe and the nucleus during elastic interaction (see Figure 19).

Figure 18. Dependence of the dipole moment modulus on the distance between the particle X and the
target nucleus Na (dot line) in the Thomson model, for n = 1, with elastic scattering and a non-zero
impact parameter.
Symmetry 2023, 15, 2182 21 of 30

In Figure 19, the blue circle shows the sum of the radii of the n-helium nucleus and the
sodium target nucleus, while the red asterisk shows the initial position of the particles of the
XHe atom. The violet and red dotted lines show graphs of the dependence of the Coulomb
potential between nHe and the nucleus and between X and the nucleus, respectively, on the
distance between nHe and the nucleus. Red, blue and green solid lines show graphs of the
dependence of the Stark potential, the total potential acting on n-helium, and the Coulomb
potential between nHe and X, respectively, on the distance between nHe and the target
nucleus. The black dotted line shows a graph of the total effective interaction potential
between XHe and the nucleus as a function of the distance between nHe and the nucleus.

Figure 19. Dependence on the distance between nHe and the nucleus during elastic interaction,
for n = 1: of the Coulomb potential between nHe and the nucleus (purple dotted line), between X
and the nucleus (red dotted line), and between nHe and X (green solid line), of the Stark potential
(red solid line), of the total potential acting on nHe (blue solid line), and of the total effective potential
(black dotted line).

In Figure 19, one can see that the dark matter atom is weakly polarized, the Coulomb
force between nHe and X remains approximately constant, nHe is repelled from the target
nucleus stronger than X is attracted to it, and the «dark atom» experiences elastic scattering.
The result of inelastic interaction with a non-zero impact parameter can be seen in
Figure 20, which shows the trajectories of motion of nHe and X particles in the XZ plane
during interaction with the Na nucleus.
In Figure 20, the blue dots and the red dotted line show the trajectories of n-helium
and the X particle during inelastic interaction in the XZ plane, respectively. The green
diamond shows the initial position of the dark matter atom particles. The blue diamond
shows the final position of the X particle. The black circle shows the origin of the coordinate
system. From Figure 20, it is clear that as a result of inelastic scattering, the XHe atom is
destroyed and the X particle remains in the target nucleus, while the nHe nucleus flies
further in the original direction.
Symmetry 2023, 15, 2182 22 of 30

Figure 20. Trajectories of motion of the nHe (blue dots) and X particles (red dotted line), for n = 1,
in the XZ plane with a non-zero impact parameter for inelastic interaction.

The impact parameter and the minimum distance between the α-particle and the target
nucleus, corresponding to the interaction shown in Figure 20, are equal to
β = 0.3 · 10−13 cm , Rαmin = 1.8 · 10−15 cm.
Since, as a result of inelastic scattering and destruction of a «dark atom», the particle
X oscillates around the target nucleus, and the helium nucleus flies away to infinity,
the calculation of the length of the dipole moment, which is defined as the difference in
the radius vectors of the particles of the «dark atom», over the entire interval is unphysical.
Therefore, we present here the results calculated before the moment of nHe leaving the
target nucleus: the maximum value of the dipole moment is δmax = 5.59 · 10−13 cm; and
the average value of the dipole moment is δmean = 5.41 · 10−14 cm.
It is also possible to plot the dependence of various potentials corresponding to the
forces acting between particles in the XHe–nucleus system, depending on the distance
between nHe and the nucleus during inelastic interaction (see Figure 21). The potential
graphs in Figure 21 are labeled in the same way as in Figure 19.
From Figure 21, it is clear that particles of the XHe «dark atom» fall into the target
nucleus. Afterwards, the X particle remains in the nucleus; this can be seen from the
characteristic hole in the Coulomb potential between the X particle and the target nucleus.
A dark matter atom is inelastically scattered due to the absence of an explicitly specified
quantum–mechanical connection between the particles of the «dark atom» in the Thomson
model. Therefore, despite the fact that the Stark and Coulomb force repels n-helium from
the target nucleus, the particle X is attracted to it essentially as an independent particle.
Consequently, it is possible to improve Thomson’s model with the Stark force, getting
rid of its shortcomings, only by explicitly adding to this model the quantum mechanical
connection between the particles of the «dark atom».
Symmetry 2023, 15, 2182 23 of 30

Figure 21. Dependence on the distance between nHe and the nucleus during inelastic interaction,
for n = 1: of the Coulomb potential between nHe and the nucleus (purple dotted line), between X
and the nucleus (red dotted line), and between nHe and X (green solid line), of the Stark potential
(red solid line), of the total potential acting on nHe (blue solid line), and of the total effective potential
(black dotted line).

5. Approach of Reconstructing of Interaction Potentials in the XHe-Nucleus System

To reconstruct the effective potential of interaction between XHe and the nucleus
of matter, it is necessary to take into account all types of interaction existing in a given
three-body system. We must take into account the electromagnetic interaction between the
XHe «dark atom» and the nucleus as well as the nuclear interaction between helium and
the nucleus of the substance. And it is also necessary to take into account the Stark effect
that occurs in an external, alternating electric field created by the nucleus of the substance,
which leads to the polarization of XHe.

5.1. Electric potential of X-helium

To obtain the electric potential, φ, created by the XHe «dark atom» at a distance r from
the center of the nHe nucleus, in which we place the nucleus of the substance, we need to
e−r/r0
solve the self-consistent Poisson equation for the test wave function ψ = √ 3/2 , where r0
πr0
is a free parameter:
1 ZX e−2r/r0
 
00
(φr ) = −4πe n p + , (41)
r πr03
where en p is the charge density of nHe:

 eZα

 for r < RnHe ,
4 3

en p = πR (42)


 3 nHe
0 for r > RnHe .
Symmetry 2023, 15, 2182 24 of 30

Thus, by solving this equation, we obtained the potential created by the XHe atom,
consisting of a finite nHe nucleus and an X particle, taking into account the screening effect
of the X particle on the nHe nucleus:

1 1
 
−eZX e

 − 2r/r 0 + for r > RnHe ,
 r r
 0



1 1 eZX
 
φ = −eZX e−2r/r0 + + + (43)

 r0 r r
eZα 3 r2

  

+ − 2 for r < RnHe .


RnHe 2 2RnHe

The potential energy of electrical interaction between XHe and the nucleus is equal to
e
UXHe = eZ A φ, (44)

where Z A is the charge number of the nucleus of the substance.

5.2. Results of Reconstructing the Effective Interaction Potential in the XHe–Nucleus System with
a Nuclear Force of the Woods–Saxon Type
Using the results of calculations, we created graphs of the dependence of the nuclear
potential of the Woods–Saxon type, UXHe e (see Formula (44)), the Stark potential (recovered
by calculating the dipole moment, see Formulas (18) and (19)), and the total interaction
potential of a dark matter atom with the nucleus of matter on the distance between nHe
and the nucleus of matter for n = 1, that is, for the OHe atom. The sodium nucleus was
considered as the outer nucleus (see Figure 22).
Potentials are restored at the folllowing intervals: [R NaHe ; 20R He ], where R He = 2.5 fm
is the root-mean-square radius of the helium nucleus, and R NaHe = 5.7 fm is the sum of
the root mean square radii of the helium nucleus and the Na nucleus, respectively.

Figure 22. Dependence on the distance between He and the Na nucleus of the nuclear potential of
e
the Woods–Saxon type (green dotted line), UXHe (blue dotted line), of the Stark potential (gray dotted
line) and of the total potential between OHe and the Na nucleus (red dotted line).

Figure 22 contains graphs with green, blue, gray and red dotted lines, showing the
e
nuclear potential of the Woods–Saxon type, UXHe , the Stark potential, and the total potential,
respectively, depending on the distance between the helium nucleus of the OHe atom and
the nucleus of matter. The red circle shows the sum of the root-mean-square radii of the
helium nucleus and the nucleus of matter. In this figure, the total potential qualitatively
repeats the expected shape of the effective potential of interaction between OHe and the
Symmetry 2023, 15, 2182 25 of 30

nucleus of a substance. The total potential has a barrier, and you can also see the bound
state well of the O-helium «dark atom» with a sodium nucleus with an energy of about
0.26 MeV.
Figure 23 plots the dependence of the dipole moment on the distance between the
O−− particle and the target nucleus Na. From the figure, it can be seen that the closer the
dark matter atom is to the nucleus of the substance, the greater the polarization, which is
consistent with theory.
The maximum value of the dipole moment OHe when interacting with a sodium
nucleus is δmax = 4.96 · 10−12 cm. The average value of the dipole moment OHe during
interaction with a sodium nucleus is δmean = 1.05 · 10−13 cm.

Figure 23. Dependence of the magnitude of the dipole moment on the distance between the O−−
particle and the Na nucleus (dot line) when determining the nuclear interaction by a nuclear force of
the Woods–Saxon type.

5.3. Results of Reconstructing the Effective Interaction Potential in the XHe–Nucleus System with
a Nuclear Force Taking into Account the Non-Point Nature of Interacting Nuclei
Using the results obtained in the article [36], in our semiclassical approach to calculat-
ing the effective potential of interaction of XHe with the nucleus of matter, we replaced
the Woods–Saxon nuclear potential, which generally does not take into account the finite
sizes of interacting nuclei, with the nuclear potential, which is calculated according to
Formula (23) from the article [36]. When deriving this formula, the densities of nucleons of
interacting nuclei and the density of the binary nuclear system are taken into account. Due
to the small overlap of nuclei, the spin interaction is neglected, and both nuclei are consid-
ered as spherical. As a result, the nuclear potential of interaction between helium and the
sodium nucleus was calculated numerically using the previously mentioned formula. Next,
using the difference scheme, the corresponding nuclear force was calculated. After that,
using this nuclear force, we obtained the values of the dipole moment using Formula (18)
of this work, with the help of which, in turn, the Stark potential was calculated.
Using the results of calculations, graphs were constructed of the dependence of the
nuclear potential, taking into account the non-pointwise nature of interacting nuclei, UXHee ,
the Stark potential, and the total potential of interaction of an OHe atom with the nucleus
of a substance on the distance between He and the nucleus of a substance (see Figure 24).
The sodium nucleus was considered as the outer nucleus.
Symmetry 2023, 15, 2182 26 of 30

Figure 24. Dependence on the distance between He and the Na nucleus of the nuclear potential
e
taking into account the non−point nature of interacting nuclei (green dotted line), UXHe (blue dotted
line), of the Stark potential (gray dotted line) and of the total potential between OHe and the Na
nucleus (red dotted line).

In Figure 24 the green, blue, gray and red dotted lines show graphs of the nuclear
potential taking into account the non-pointwise nature of interacting nuclei, UXHe e , of
the Stark potential and of the total potential, respectively, depending on the distance
between the helium nucleus of the «dark atom» and the nucleus of matter. The red circle
shows the sum of the radii of the helium nucleus and the nucleus of matter. In this figure,
the total interaction potential has a well with a depth of about 0.85 MeV, but the shape of
the effective interaction potential differs from the expected one.
Within the framework of the parameters uncertainty of nuclear physics, the binding
energy in the OHe-Na system can fall within the 2–4 keV range [1,30]. This is a rather subtle
phenomenon that may have been lost due to our semiclassical approach and therefore
leading to a not quite accurate calculation of the dipole moment and, consequently, the Stark
effect. Therefore, to improve the accuracy of the results of our calculation of the effective
interaction potential, it is necessary to solve the Schrödinger equation for helium in the
OHe–nucleus system in order to calculate the polarization of the «dark atom» using a
quantum mechanical method and, thus, more accurately calculate the Stark potential.
Figure 25 plots the dependence of the dipole moment on the distance between the
O−− particle and the target nucleus Na. This figure also shows that the OHe polarization
increases with decreasing distance between the dark matter atom and the nucleus of
substance, which is consistent with theory.
The maximum value of the dipole moment for interaction with the sodium nucleus is
δmax = 1.77 · 10−12 cm. The average value of the dipole moment for interaction with the
sodium nucleus is δmean = 6.41 · 10−14 cm.
Symmetry 2023, 15, 2182 27 of 30

Figure 25. Dependence of the magnitude of the dipole moment on the distance between the O−−
particle and the Na nucleus (dot line) when determining the nuclear interaction by the nuclear force,
taking into account the non−pointwise nature of the interacting nuclei.

6. Conclusions
The lack of positive results of underground searches for WIMPs, as well as some
problems regarding a simple dark matter description of the structure and evolution of
galaxies [37,38] (as well as the data from GAIA DR3 catalogue), may posit the question of
whether dark matter does exist at all and whether its physics and effects are worth studying.
We can defend our position as follows: the Standard Model (SM) of elementary particles is
incomplete and involves its well-motivated extensions beyond the Standard Model (BSM).
Stable particles and forms of matter inevitably follow in the BSM models from the extension
of the SM symmetry. The set of data of precision cosmology involves dark matter as the
necessary element of the now standard cosmological paradigm. The problems related to
the direct dark matter search or description of the galactic structure favor in our opinion
more sophisticated but still physically well-motivated dark matter models. Here, we use
such types of models, which are linked to the composite Higgs solution of the SM problem
of Higgs boson mass divergence, to approach a solution for the puzzles of direct dark
matter searches.
The study focuses on investigating the hypothesis of composite dark matter, where
stable particles with a charge of −2n combine with helium nuclei to form neutral atom-
like states called XHe «dark atoms». These X-helium states can interact with ordinary
matter nuclei, and the specifics of this interaction can provide explanations for certain
experimental observations. It can be easily shown that in spite of their strong (nuclear)
cross-section, XHe–XHe interactions do not lead to dissipation. Their number density
is determined by the observed dark matter density and is so low that XHe gas becomes
collisionless in the early universe soon after XHe is formed and on the scale of galaxies.
Specifics of the formation of XHe and the dissipation of its density fluctuations leads dark
atom cosmology to a Warmer than Cold Dark Matter scenario of structure formation, which
needs special study but qualitatively is not excluded by the data of the precision cosmology
(see e.g., ref. [15] for a review and details).
For the XHe dark matter atom model to be viable, it is necessary for repulsive inter-
action to occur at some distance between XHe and the nucleus. Solving this problem is
crucial for the continued existence of the XHe hypothesis [23]. To prevent the excessive
production of anomalous isotopes, it is assumed that there will be a barrier in the effective
interaction potential between XHe and the nucleus, preventing fusion with the nucleus of
matter. This problem is formulated as a three-body problem and lacks an exact analytical
Symmetry 2023, 15, 2182 28 of 30

solution. Therefore, this study proposes a numerical approach to describe this interaction.
The goal is to sequentially construct a numerical model that can reconstruct the shape of
the corresponding effective potential. In this work, two semiclassical approaches are used:
the approach of reconstructing particle trajectories, which includes the Bohr model and the
Thomson model, and the approach of reconstructing the potential. These models describe
a system of three charged particles interacting through Coulomb and nuclear forces.
In the Bohr atom approximation, elastic interaction predominates, and the effective
potential of interaction between OHe and the nucleus qualitatively coincides with the
theoretically expected one: that is, there is a Coulomb barrier and potential well. However,
there are drawbacks in this model, such as the absence of an explicit specification of the
Coulomb force between helium and O−− and the fixed Bohr orbit of rotation for He in
the OHe atom, which prevents automatic polarization due to the Stark effect. On the
other hand, the Thomson model of the atom solves these issues by considering the helium
nucleus as a charged ball within which the O−− particle can oscillate. Additionally, the case
of −2 charged particles is just one possibility, as the X particles being considered can have
a charge of −2n and form a «dark atom» with n nuclei 4 He, which themselves are Thomson
atoms starting from n = 2. When modeling in the Thomson atom approximation, two
distinct cases can be observed for n = 1 regardless of the impact parameter: elastic interac-
tion and inelastic interaction when the O−− particle collides with the nucleus. In elastic
scattering, the direction of motion for OHe particles changes to the opposite direction,
and the O-helium atom experiences greater polarization as it approaches the target nucleus.
In inelastic scattering, the OHe atom is destroyed, and the O−− particle remains in the
target nucleus while the helium nucleus continues flying in its original direction.
In the Thomson model, the polarization of the «dark atom» occurs automatically due to
the Stark effect, while in the Bohr model, it had to be manually included. However, the Bohr
model preserves the quantum–mechanical connection between the «dark atom» particles,
which is represented by the fixed Bohr orbit of the α-particle. On the contrary, in Thomson’s
model, this quantum mechanical connection is lost. But at the same time, in Bohr’s model,
this connection does not allow automatic polarization of the «dark atom». Due to the
insufficient quantum mechanical connection between the particles of the «dark atom»
in Thomson’s model, the dark matter atom scatters inelastically. Despite the repulsion
of helium from the nucleus by the Stark and Coulomb forces, O−− is attracted to it as
an independent particle. This means that polarization cannot be manually introduced;
however, in this case, the quantum–mechanical connection in XHe is lost. Alternatively,
this connection can be preserved, but then the length of the dipole moment needs to
be calculated in an unnatural way, which affects the accuracy of the results. Therefore,
Thomson’s model requires a stricter quantum–mechanical connection between helium and
the X particle.
In the semiclassical approximation of the potential restoration approach, the effective
potential of interaction between XHe and the nucleus of a substance exhibits the anticipated
features: a Coulomb barrier and a bound state well in the XHe–nucleus system. However,
when considering the finite dimensions of the nucleus in the nuclear force for the Na
nucleus, the shape of the effective interaction potential deviates significantly from what
was expected. The existence of binding energy in the OHe-Na system is a delicate effect
that could have been lost since the Stark effect was calculated in a semiclassical way, which
surely reduced the accuracy of the results obtained.
From the analysis of the length of the dipole moment, it is clear that the greater
the polarization of a «dark atom», the closer it is to the nucleus of a substance, which is
consistent with the theory. And the maximum possible value of the length of the dipole
moment δmax of a «dark atom» when interacting with the nucleus Na is equal to about
10−12 cm.
To enhance the precision of determining the effective interaction potential, it is crucial
to adopt a quantum mechanical method in the future. This entails solving the Schrödinger
equation for helium in the XHe-nucleus system to compute the polarization of the dark
Symmetry 2023, 15, 2182 29 of 30

matter atom in a quantum mechanical manner. By doing so, a more accurate evaluation of
the Stark potential can be achieved.
Thus, in the future, it is planned to solve the Schrodinger equations in the XHe-nucleus
system. Then, it is planned to calculate the dipole moments of the polarized XHe atom
using the reconstructed wave functions of helium in an isolated «dark atom» and in the
XHe-nucleus system. Afterwards, we plan to restore the Stark potential and construct
an effective interaction potential of the XHe–nucleus system, in which, in addition to
the Stark potential, the nuclear potential and the electromagnetic potential are also taken
into account.

Author Contributions: Article by T.B., M.K. and A.M. The authors contributed equally to this work.
All authors have read and agreed to the published version of the manuscript.
Funding: The work by M.K. and A.M. was performed with the financial support provided by the
Russian Ministry of Science and Higher Education, project “Fundamental and applied research of
cosmic rays”, No. FSWU-2023-0068.
Data Availability Statement: No new data were created or analyzed in this study. Data sharing is
not applicable to this article.
Conflicts of Interest: The authors declare no conflict of interest.

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