Dark Atoms of Nuclear Interacting Dark Matter

Abstract

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.

1. Introduction

The modern cosmological paradigm involves dark matter (DM) as the dominant (more than $85\%$) component of the matter content of the Universe. The formation of cosmological large-scale structures, the behavior of galaxies, gravitational lensing, the anisotropy of the cosmic microwave background and other astrophysical observations confirm its existence. However, the nature, dynamic characteristics and possible observable manifestations of the DM remain unknown. Their explanation would definitely lead beyond the standard models of fundamental interactions, either by modification of gravity or by extension of the Standard Model (SM) of electroweak and strong interactions of elementary particles. In the latter case, the non-baryonic nature of DM implies the existence of new stable forms of nonrelativistic matter, reflecting stability of the DM particles. At the particle level, the stability of DM particles assumes that they possess new conserved charges, which SM particles do not possess, reflecting new strict symmetry that extends the SM symmetry.

There are a variety of ways to solve this problem in models describing physics beyond the SM (see [1,2,3,4,5,6,7]). In this article, we will focus on the dark atom scenario of dark matter and outline its advantages and problems. A dark atom consists of the new heavy particle $X^{-2n}$ (without or strongly suppressed QCD interaction) with even negative charge and n nuclei of ${}^4\mathit{He}$, bound by ordinary electromagnetic Coulomb force. This scenario does not involve new physics, with the exception of $X^{-2n}$, and reduces the effects of dark atoms on the nuclear interaction of their nuclear shells.

Multiple charged stable particles arise in several possible extensions of the Standard Model. One of them, namely, the model of a minimal walking technicolor (WTC) [8,9,10,11,12,13], is considered in this article. This approach becomes of special interest in the lack of positive evidence for supersymmetry (SUSY) at the energies of the Large Hadron Collider (LHC). If the SUSY energy scale corresponds to much higher energies than the electroweak scale, the problems of the Standard Model (divergence of the Higgs boson mass and the origin of the electroweak energy scale) may need a non-supersymmetric solution, which may be provided by models of a composite Higgs boson. In the WTC model, the existence of additional heavy fermions with new gauge interaction is assumed, and the Higgs boson physics is formulated in terms of a single scalar doublet describing the Higgs boson as a composite particle. This model similarly leads to a new approach to considering dark matter, revealing its composite nature.

The topology of the electroweak gauge group $SU\left(2\right)$ leads to the existence of a set of different vacua. The saddle point on the top of the potential barrier corresponds to a nontrivial solution of the field equations, and this unstable classical solution is called “sphaleron”. At high temperatures in the early Universe, it is possible to transit from one topological vacuum to another close one by a sphaleron process, changing the Chern–Simons number by unity. In such transitions, laws of baryon and lepton numbers, conservation should be violated, ensuring the observable baryon asymmetry of the Universe [14]. It enables one to balance the baryon asymmetry and an excess of dark matter particles over the corresponding antiparticles.

Because the potential barrier is high enough, ($E_{Sph}\approx 9.1\mathrm{TeV}$), sphaleron transitions have not been observed yet. Using the data of collider experiments and cosmic rays observations [15,16], only the limits on the sphaleron rate could be set. The theoretical consideration shows that the rate of this process strongly depends on the temperature (see review [17]).

Section 2 considers the cosmological consequences of sphaleron transitions. In the first Section 2.1, a thermodynamic approach is described. It was used to consider the minimal WTC model (Section 2.2). The obtained dependence of the ratio of dark and baryonic matter on the model parameters allows one to find the constraints on the masses of new heavy particles, which are consistent with the experimental data [18]. A small review of the study of the sphaleron transition rate is presented in Section 2.3.

Section 3 briefly discusses the dark atom formation at Big Bang Nucleosynthesis. It is highlighted in Section 3.1 that, depending on the charge of the multiple-charged constituent, the neutral bound state with helium nuclei could have either Bohr-like or Thomson-like “atomic” structure. This leads to some problems in the description of capturing light nuclei, which is described in Section 3.2. Section 3.3 considers the large-scale structure formation triggered by dark atoms.

Although numerous experiments have been conducted to directly detect dark matter particles, they have given conflicting results. Section 4 examines the potential role of X-helium atoms of composite dark matter in resolving these contradictions and advancing our understanding of how dark matter interacts with ordinary matter.

Section 4.1 discusses the hypothesis of dark atoms of X-helium as a means of explaining the various contradictory results of experiments on the direct detection of dark matter particles. This hypothesis suggests that the non-trivial nature of the interaction between dark atoms and nuclei of ordinary matter in the detector material and the resulting formation of low-energy bound states in the XHe-nucleus system can explain the positive results observed in the $DAMA/NaI$ and $DAMA/LIBRA$ experiments, in contrast to the negative results of other experiments such as $XENON100$, $LUX$, $CDMS$, etc. [7].

Section 4.2 delves into the challenges posed by the potential strong interaction of XHe atoms with matter nuclei, which could disrupt the bound state of dark atoms and lead to the formation of anomalous isotopes, the abundance of which in the environment is subject to strict experimental constraints [19]. To solve this problem, the XHe hypothesis assumes the existence of a dipole Coulomb barrier in the effective interaction potential of a dark atom with a nucleus, preventing the fusion of the $nHe$ and X particles that make up the dark atom with the nuclei of matter. This condition is vital for the existence of the X-helium hypothesis. Given the complexity of this three-body problem, an exact analytical solution is not available, necessitating the development of a numerical approach to validate the proposed scenario.

Section 4.3 presents systematic, step-by-step development of a numerical quantum-mechanical model describing the interaction of a OHe dark atom with a sodium nucleus. By incorporating necessary effects and interactions, we aim to enhance the accuracy of our model and clarify key aspects of the proposed dark atom scenario. The developed numerical quantum mechanical model reconstructs the effective interaction potential, providing a detailed characterization of the features and nuances of the interaction of dark atoms of composite dark matter with the nucleus of a substance.

2. Generation of Multiple Charged Dark Atom Constituents

2.1. Thermodynamic Approach for the Study of Sphalerons

The evolution of the early universe is a time sequence of equilibrium and non-equilibrium states of cosmological plasma. Its composition changes at different stages of the evolution as a result of phase transitions occurring in a system of interacting particles in the process of a gradual decrease in the temperature of the system. However, even under conditions of thermodynamic equilibrium, sphaleron transitions can occur, the rate of which depends on the temperature and density of the particle plasma. The consequence of these processes connecting changes in the baryon and lepton numbers of particles and their bound states is the emergence and establishment of a fixed baryon asymmetry of the universe in the process of its cooling.

Sphalerons are classical many-particle processes, so the language of thermodynamics seems adequate for their analysis, including the consideration of their consequences for cosmology [20] (see also [21,22]). Subsequently, the thermodynamic approach was generalized to the case of sphalerons in the Standard Model extensions [12,13]. The equilibrium conditions in cosmological plasma can be written down for transitions between vacuum states using the chemical potentials of any left/right particles. Then, we obtain a relationship between particle number densities and the possibility of studying the dependence of energy densities on model parameters.

Therefore, assuming the thermal equilibrium at some interval of temperatures, the density of baryon number is defined as follows:

$$B={\displaystyle \frac{6}{g{T}^2}}(n_b-n_{\overline{b}})={\displaystyle \frac{1}{3}}T\sum {\displaystyle \frac{\mu }{T}}\sigma \left({\displaystyle \frac{m}{T}}\right).$$

(1)

Here, $n_b$ and $n_{\overline{b}}$ are distributions for densities of baryons and antibaryons, and Taylor expansion was used to obtain such expression of the baryon number. Also, characters g, $\mu$ and m represent the degrees of freedom, chemical potential and mass of particular particles, correspondingly. As usual, factor ${\displaystyle \frac{1}{3}}$ is the baryon number of a quark. The known weight function for a massive particle, $\sigma$, has the form:

$$\sigma \left(z\right)=\left\{\begin{array}{c}{\displaystyle \frac{6}{4{\pi }^2}}{\int }_0^{\infty }dx{x}^2{cosh}^{-2}\left({\displaystyle \frac{1}{2}}\sqrt{x^2+z^2}\right),\mathrm{for}\mathrm{fermion};\hfill \\ {\displaystyle \frac{6}{4{\pi }^2}}{\int }_0^{\infty }dx{x}^2{sinh}^{-2}\left({\displaystyle \frac{1}{2}}\sqrt{x^2+z^2}\right),\mathrm{for}\mathrm{boson}.\hfill \end{array}\right.$$

(2)

The lepton number density, as well as the density of charge, can be written analogously. Obviously, the sign for chemical potential is the same as for the density: it is positive for a particle and negative for an antiparticle.

To reduce the number of chemical potentials for the equilibrium equations, we use the following electroweak conditions

$$\begin{array}{c}\hfill {\mu }_{iR}={\mu }_{iL}\pm {\mu }_0,\end{array}$$

(3)

$$\begin{array}{c}\hfill {\mu }_i={\mu }_j+{\mu }_W,\end{array}$$

(4)

where the indices “i” and “j” denote the weak doublet components, and ${\mu }_W$ and ${\mu }_0$ are chemical potentials of $W^{-}$ and Higgs bosons, correspondingly.

In the WTC model, additional fermions arise with new charges distinguishing them from the standard fermions. New stable (or long-living bound) states can be produced with the participation of extra fermions from WTC. Densities of corresponding quantum numbers, $TL,TB$, for technileptons and technibaryons are necessary characteristics for the kinetics of sphaleron transitions, connecting changes in baryon and lepton numbers.

The basic parameter of cosmological plasma, in which phase and sphaleron transitions occur, is the temperature of the medium. The height of the potential barrier (or the energy of the sphaleron) as well as the sphaleron rate can be estimated in the Standard Model by using the value $T_f\sim$ 100–200 GeV [23,24] and supposing that this is the freeze-out temperature of sphaleron. However, these parameters can change in various extensions, and therefore, we will consider a slightly extended temperature range, $T_f=$ 150–250 GeV, as was suggested in Refs. [12,13]. As for the Standard Model particles (excluding t-quark), the inequality ${\displaystyle \frac{m}{T_f}}\ll 1$ holds for them, so that ${\sigma }_f\approx 1$ (for fermions) and ${\sigma }_b\approx 2$ (for bosons).

In fact, the main parameter for the system of equations for chemical potentials is the ratio of temperatures: the sphaleron freeze-out temperature $T_f$ and the electroweak phase transition (EWPT) temperature $T_c$. For high temperatures, $T_f>T_c$, i.e., when sphalerons switch off before the EW phase transition occurs, the isospin neutrality condition, $I_3=0$, could be used as an additional equation. However, for the case $T_f<T_c$, the chemical potential of Higgs boson turns to zero ${\mu }_0=0$ as a consequence of a condensation process in a nonperturbative vacuum.

At zero approximation for SM extensions, we can take masses of additional particles as equal. Namely, these are masses of new fermions, techniquarks, in WTC. In addition, masses of the lowest stable bound states of techniquarks are approximately equal too. In this most-simplified case, we obviously can isolate a dominant and robust result from our equations. Corrections to this approximation can be studied; however, this analysis is cumbersome, and the masses are arguments of weight functions in equations for sphaleron transitions. Therefore, we consider the following parameterization:

$$d_i=\sigma \left({\displaystyle \frac{m_i}{T_f}}\right)-\sigma \left({\displaystyle \frac{m}{T_f^{*}}}\right),$$

(5)

where m and $T_f^{*}$ are selected for convenience reasons, $T_f$ is a sphaleron freeze-out temperature. Parameter $d_i$ should vary in the limits $[-\sigma \left({\displaystyle \frac{m}{T_f}}\right);0]$ and describe the values of “i”-particle’s mass $[m;+\infty ]$.

2.2. Balancing of Tecniparticle Excess in WTC Model

Technicolor models were originally inspired by the idea of a non-elementary composite Higgs boson (i.e., a bound state of new fermions, techniquarks), whose mass is generated by dynamic symmetry breaking. Then, the model was improved by the special choice of the running coupling behavior, resulting in a model of walking technicolor (WTC) and satisfying the restrictions imposed by Peskin–Tackeuchi parameters. Importantly, the WTC model global symmetry $SU\left(4\right)$ is broken to $SO\left(4\right)$, generating nine Goldstone bosons [8,9,10,11,25] (note, another way for the global symmetry to break the symplectic group $SU\left(4\right)\to Sp\left(4\right)$ is more economic and leads to another set of possible final states [26]). Zero mass Goldstones provide necessary longitudinal components for massive gauge EW bosons W, Z and also for new composite states—technibaryons $UU$, $UD$, $DD$ and their antiparticles.

Additional heavy fermions—techniquarks, U, D, and technileptons, N, E—belong to an adjoint representation of $SU\left(2\right)$, so the techniquarks acquire arbitrary electric charges. Additional technileptons, N and E, with some conditions for their masses, are introduced into the model to compensate for anomaly contributions. An important detail is that the WTC model contains hidden mass candidates in the form of so-called dark atoms.

As for the arbitrary electric charges, they can be parameterized in some way; it is useful to introduce, for example, an integer parameter, y, as was done in Refs. [12,13]: $q=y+1$, y, $y-1$ and ${\displaystyle \frac{1}{2}}(-3y+1)$, ${\displaystyle \frac{1}{2}}(-3y-1)$ for $UU,UD,DD$ and $N,E$, accordingly: a special case with $y=1$ was also analyzed in these papers. Notice, in the WTC, technibaryons form the electroweak triplet ($UU$,$UD$,$DD$) with isospin projections $(1,0,-1)$, and left leptons make a doublet (N,E).

In this scenario, we consider the lightest techniparticles as stable, and the dark atoms constituting hidden mass are bound states, namely: the $(-2n)$-charged (anti)technibaryon $U{U}^{-2n}$ and (anti)technileptons ${(N/E)}^{-2n}$ can represent the cores of X-helium dark atoms $X^{-2n}{\left(H{e}^{+2}\right)}_n$, which should have masses $\gtrsim 1\mathrm{TeV}$.

This estimation follows on from the experiments of ATLAS [18] in the search for multicharged particles, i.e., we can define a lower limit for the techniparticles masses. Also, we suppose that $m_U<m_D$. As for technileptons, generally speaking, their masses are not fixed; furthermore, the characteristics of the lightest lepton flavor depend on the charge parameter y. Importantly, for this flavor, the known no-go theorem should be satisfied [7].

Densities of the standard baryon and lepton numbers are defined in the usual way, as is described in Refs. [12,13,20]:

$$\begin{array}{cc}\hfill & \begin{array}{c}\hfill B=(10+2{\sigma }_t){\mu }_{uL}+6{\mu }_W,\end{array}\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill & \begin{array}{c}\hfill L=4\mu +6{\mu }_W.\end{array}\hfill \end{array}$$

(7)

Densities of technibaryon and technilepton numbers are distinguished from the standard ones and defined as

$$\begin{array}{cc}\hfill & \begin{array}{c}\hfill TB={\displaystyle \frac{2}{3}}({\sigma }_{UU}{\mu }_{UU}+{\sigma }_{UD}{\mu }_{UD}+{\sigma }_{DD}{\mu }_{DD}),\end{array}\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill & \begin{array}{c}\hfill TL={\sigma }_E({\mu }_{EL}+{\mu }_{ER})+{\sigma }_N({\mu }_{NL}+{\mu }_{NR}).\end{array}\hfill \end{array}$$

(9)

Here, we use chemical potentials ${\mu }_{UU},{\mu }_{UD},{\mu }_{DD}$ for technibaryons and ${\mu }_{NL/R},{\mu }_{EL/R}$ for left/right technileptons.

Now, we can write an equation of the electroneutrality for plasma using the y-parameterization for the multicharged particles:

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill 0=& {\displaystyle \frac{2}{3}}·3·3({\mu }_{uL}+{\mu }_{uR})-{\displaystyle \frac{1}{3}}·3·3({\mu }_{dL}+{\mu }_{dR})-{\displaystyle \frac{1}{3}}({\mu }_{eL}+{\mu }_{eR})+\hfill \\ \hfill & +(y+1){\sigma }_{UU}{\mu }_{UU}+y{\sigma }_{UD}{\mu }_{UD}+(y-1){\sigma }_{DD}{\mu }_{DD}+\hfill \\ \hfill & +{\displaystyle \frac{-3y+1}{2}}{\sigma }_N({\mu }_{NL}+{\mu }_{NR})+{\displaystyle \frac{-3y-1}{2}}{\sigma }_E({\mu }_{EL}+{\mu }_{ER})-4{\mu }_W-2{\mu }_m.\hfill \end{array}\hfill \end{array}$$

(10)

Analogously to Refs. [12,13], we write an equation for sphaleron transitions and add an important condition of neutrality of the total isospin projection in the high-temperature case, $T>T_c$:

$$3({\mu }_{uL}+2{\mu }_{dL})+\mu +{\displaystyle \frac{1}{2}}{\mu }_{UU}+{\mu }_{DD}+{\mu }_{NL}=0,$$

(11)

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill 0=& {\displaystyle \frac{1}{2}}·3·3·({\mu }_{uL}-{\mu }_{dL})+{\displaystyle \frac{1}{2}}·3·3·({\mu }_{iL}-{\mu }_{eL})+\hfill \\ \hfill & +{\sigma }_{UU}{\mu }_{UU}-{\sigma }_{DD}{\mu }_{DD}+{\displaystyle \frac{1}{2}}{\sigma }_N{\mu }_{NL}-{\displaystyle \frac{1}{2}}{\sigma }_E{\mu }_{EL}-4{\mu }_W-{\mu }_W.\hfill \end{array}\hfill \end{array}$$

(12)

Next, a few conditions for chemical potentials could be introduced. Namely, as in Ref. [12], the electroweak decay equations are

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\mu }_{UD}={\mu }_{UU}+{\mu }_W,\hfill \\ \hfill & {\mu }_{DD}={\mu }_{UU}+2{\mu }_W.\hfill \end{array}\end{array}$$

(13)

For the lightest technilepton, there is an electroweak condition, which is similar to the standard ones. As for the chemical potential for the Higgs boson, it has zero value and does not depend on the sphaleron freeze-out temperature, i.e.,

$${\mu }_0=0.$$

(14)

The reason for this choice is obvious: in Equations (11) and (13), we define the chemical potential of the bound state as a sum of potentials for its constituents. In the scenario considered, the Higgs boson is treated as the bound state ${\displaystyle \frac{1}{\sqrt{2}}}(U\overline{U}+D\overline{D})$, so its chemical potential should be zero due to opposite signs in the chemical potentials of the particle and antiparticle. This is different from the definition in [12,13]. Note, if the sphaleron freeze-out takes place before the EWPT, one more equation should be added to the above equations.

2.2.1. Sphaleron Freezing out before the EWPT

Therefore, having the chemical potentials for all components of the plasma for the case when sphalerons freeze out before the EWPT, the system of equations can be solved with respect to two convenient variables—density ratios:

$${\displaystyle \frac{TB}{B}}=-{\displaystyle \frac{{\sigma }_{UU}(3y{\sigma }_E-1)}{3y({\sigma }_{UU}+3{\sigma }_E)}}\left({\displaystyle \frac{L}{B}}+{\displaystyle \frac{9y{\sigma }_E+1}{3y{\sigma }_E-1}}\right),$$

(15)

$${\displaystyle \frac{TL}{B}}=-{\displaystyle \frac{{\sigma }_E(y{\sigma }_{UU}+1)}{y({\sigma }_{UU}+3{\sigma }_E)}}\left({\displaystyle \frac{L}{B}}+{\displaystyle \frac{3y{\sigma }_{UU}-1}{y{\sigma }_{UU}+1}}\right),$$

(16)

where the masses of technibosons and technileptons are considered as equal, $m_{UU}=m_{UD}=m_{DD}$ and $m_N=m_E$. The structure of the solutions found is similar

$${\displaystyle \frac{TB}{B}},{\displaystyle \frac{TL}{B}}=-\alpha \left({\displaystyle \frac{L}{B}}+\beta \right).$$

(17)

Therefore, these expressions can be combined into the ratio of dark and baryonic matter densities:

$${\displaystyle \frac{{\Omega }_{DM}}{{\Omega }_b}}\approx {\displaystyle \frac{{\Omega }_{UU}}{{\Omega }_b}}+{\displaystyle \frac{{\Omega }_L}{{\Omega }_b}}={\displaystyle \frac{3m_{UU}}{2m_p}}\left|{\displaystyle \frac{TB}{B}}\right|+{\displaystyle \frac{3m_{N/E}}{m_p}}\left|{\displaystyle \frac{TL}{B}}\right|.$$

(18)

Without the loss of generality, we can assume that the masses of technibaryons and technyleptons are proportional to each other, in particular, $m_{N/E}={\displaystyle \frac{m_{UU}}{2}}$. This ratio provides some suppression of the technibaryon density. Then, considering the case of a large mass of additional particles in this model, the prefactors in the Equations (15) and (16) can be written as follows:

$$\begin{array}{cc}\hfill & \begin{array}{c}\hfill {\alpha }_{TB}={\displaystyle \frac{{\sigma }_{UU}(3y{\sigma }_E-1)}{3y({\sigma }_{UU}+3{\sigma }_E)}}\sim {\displaystyle \frac{{\sigma }_{UU}}{{\sigma }_E}},\end{array}\hfill \end{array}$$

(19)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\alpha }_{TL}={\displaystyle \frac{{\sigma }_E(y{\sigma }_{UU}+1)}{y({\sigma }_{UU}+3{\sigma }_E)}}\sim {\displaystyle \frac{{\sigma }_E}{{\sigma }_E}}=1,\end{array}\end{array}$$

(20)

and the ratio ${\displaystyle \frac{{\sigma }_{UU}}{{\sigma }_E}}$ decreases exponentially.

Depending on the charge parameter, the density ratios (${\displaystyle \frac{{\Omega }_{UU}}{{\Omega }_b}}$ and ${\displaystyle \frac{{\Omega }_L}{{\Omega }_b}}$) behave hyperbolically, as shown in Figure 1. Note that this dependence results from the condition for the Higgs boson chemical potential.

Due to the presence of the factors $(3y{\sigma }_E-1)$ and $(y{\sigma }_{UU}+1)$ in the Equations (15) and (16), the sign of the density ratios can change to the opposite. This means that the suggested composition of dark atoms depends on the model parameters. The Line break in the left panel of Figure 1 shows at what values the excess of technibaryons is transformed into an excess of antitechnibaryons. The “critical” mass values at which this occurs for several special cases are listed in

Table 1. The critical values of total mass assuming $m=m_{N/E}={\displaystyle \frac{m_{UU}}{2}}$.

y	${\mathit{\sigma }}_{\mathit{f}}^{\mathbf{crit}}=\frac{1}{3\mathit{y}}$		${\mathit{m}}^{\mathbf{crit}}$, GeV
		${\mathit{T}}_{\mathit{f}}=\mathbf{250}\mathbf{GeV}$	${\mathit{T}}_{\mathit{f}}=\mathbf{200}\mathbf{GeV}$	${\mathit{T}}_{\mathit{f}}=\mathbf{150}\mathbf{GeV}$
3	${\displaystyle \frac{1}{9}}$	≈1141	≈913	≈684
5	${\displaystyle \frac{1}{15}}$	≈1311	≈1048	≈786

.

If the charge parameter equals unity, i.e., $y=1$, the critical value of technibaryon mass is sufficiently low, so we come to the following inequality

$${\sigma }_f^{\max }\left({\displaystyle \frac{m_f^{\min }}{T_f}}\right)={\sigma }_f\left({\displaystyle \frac{1000}{250}}\right)\approx 0.167<{\displaystyle \frac{1}{3y}}={\displaystyle \frac{1}{3}}.$$

It means that the changing of the sign of (anti) techniparticles excess is absent in this case.

Notice some physically unreasonable possibility of changing the sign of the technilepton number density for the case $y<0$. Quantitatively, the maximum value of the bosonic weight function is

$${\sigma }_b^{\max }={\sigma }_b\left({\displaystyle \frac{2000}{250}}\right)\approx 0.007.$$

Then, at $y\approx -{\displaystyle \frac{1}{{\sigma }_b^{\max }}}\approx -138$, the sign of density is changed. Obviously, this value of y is physically unacceptable.

In this consideration, the ratio ${\displaystyle \frac{L}{B}}$ can vary in very broad limits, practically up to infinity. Such behavior should significantly affect other physical parameters of the scenario, for instance, mass dependence of the densities ratio, as seen in Figure 2, for the specific case $y=3$. Here, the ATLAS experimental limit [18] is denoted by the vertical green lines, the observed value of the density ratio [27] is depicted by the horizontal line.

As is seen from (18), the ratio ${\displaystyle \frac{L}{B}}$ comes into this equation linearly, which means the possibility of some unwanted overproduction of techniparticles during sphaleron processes. However, experiments significantly restrict the parameter space value, so only a narrow region remains permitted, and this region is marked as white and shown in Figure 3. This area is near vertical lines in Figure 3, indicating points at which the sign of the number densities ratio changes for various ${\displaystyle \frac{L}{B}}$ values. The forbidden areas are marked with grey color; these regions should be excluded because of the following reasons:

(1)

—The overproduction of both types of techniparticles;

(2)

—The overproduction of technileptons;

(3)

—The overproduction of technibaryons;

(4)

—The overproduction of the anomalous isotopes, which occur as a consequence of the excess positively charged techniparticles.

The techniparticle overproduction can also occur in the scenario of multicharged ($y\le 3$) and sufficiently light particles, $m<m^{\mathrm{crit}}$. Prohibition of this unwanted effect allows setting a lower limit for mass.

Figure 3. Allowed and forbidden regions for small values of charge parameter y.

Figure 3. Allowed and forbidden regions for small values of charge parameter y.

Thus, we can conclude that dark atom composition is defined by the charge parameter value, y. In more detail, for the permitted region, we have:

• If $y>0$, there are two forms of X-helium; hidden mass consists mainly of technilepton dark atoms, ${(N/E)}^{{\displaystyle \frac{-3y\pm 1}{2}}}{{(}^{4}H{e}^{+2})}_{{\displaystyle \frac{-3y\pm 1}{4}}}$, and there is also the (exponentially suppressed) technibaryons density, which is provided by ${\left(\overline{U}\overline{U}\right)}^{y+1}{{(}^{4}H{e}^{+2})}_{{\displaystyle \frac{y+1}{2}}}$ states;
• In the special case $y=-1$, only one dark atom type arises—technileptonic O-helium $\overline{E}He$. In addition, technibaryons occur in the WIMP form, $\overline{U}\overline{U}$, but their density is suppressed.
• For $y<-1$, we found that electric charges of technileptons and technibaryons have different signs; as a consequence, a much more complex set of techniquark bound states is possible. In this case, the allowed states are located near the right boundary of the permitted region; the hidden mass density is almost saturated by the WIMP-like states ${\left(\overline{U}\overline{U}\right)}_m{\left(\overline{N}\right)}_n$, where m and n fulfill the following equation:

  $$-2m(y+1)-n(-3y\pm 1)=0.$$

  (21)
  Nevertheless, at the left boundary, there arise two additional forms of X-helium states generated due to increasing the technileptons density. Namely, dark atoms with technileptonic ${\overline{N}}^{-2r}$ and mixed ${\overline{N}}^{-2r}{\left(\overline{U}\overline{U}\right)}^{+2s}$ cores can emerge and contribute to the total hidden mass relic abundance.

This scenario is implemented assuming the fulfillment of the relation between the masses $m_{UU}\ge m_{N/E}$, which entails a corresponding change in the boundaries of the allowed area. To estimate this area’s dynamics, it is useful to consider the following function

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \Delta =\left|{\displaystyle \frac{L}{B}}\left({\displaystyle \frac{TB}{B}}=0\right)-{\displaystyle \frac{L}{B}}\left({\displaystyle \frac{TL}{B}}=0\right)\right|=\left|{\beta }_{TB}-{\beta }_{TL}\right|\\ \hfill \approx 4\left|{\displaystyle \frac{y({\sigma }_{UU}+3{\sigma }_E)}{(3y{\sigma }_E-1)(y{\sigma }_{UU}+1)}}\right|\end{array}\end{array}$$

(22)

An increase in this function unambiguously corresponds to an extension of the allowed area.

For example, assuming equal masses $m_{UU}=m_{N/E}$, we can nearly avoid suppression of the DM technibaryon component. Dependencies of the technibaryon density ratios on the mass are shown in Figure 4, and they are plotted for different values of ${\displaystyle \frac{L}{B}}$ for a fixed temperature, $T=200\mathrm{GeV}$. Clearly, these curves behave analogously to the technileptonic density ratio. In the framework of the above assumption, the dark atoms with a technibaryonic core should be an important component of the DM, but at the same time, the danger of overproduction of such bound states increases significantly.

Independent of the sign of the charge parameter y, the removal of the technibaryon suppression ensures the emergence of the WIMPs. For $y<0$, we obtain the exact same description of the DM in the form of WIMPs. In the case of $y>0$, some part of anomalous isotopes also transforms into WIMPs at (4)-type areas.

The general dependence of density ratios on the mass difference $\Delta m=m_{UU}-m_{N/E}$ is presented in Figure 5. Note, however, that to avoid the overproduction of DM, in most cases, we need some fine-tuning of the model parameters.

From the above suggestions about masses of techniparticles, it follows that the maximum value of the bosonic weight function is ${\sigma }_b^{\max }={\sigma }_b\left({\displaystyle \frac{1000}{250}}\right)\approx 0.171$. Therefore, taking the charge parameter $y<-{\displaystyle \frac{1}{{\sigma }_b^{\max }}}\approx -5.8$ and varying the techniparticle mass with this y, we find the changing of the technilepton number sign. Nevertheless, for the fulfillment of the non-go theorem, it is necessary to analyze this effect for $y\le -7$ more carefully.

Considering the special case $y=-7$, $m_{UU}<m_{\mathrm{crit}}$, in Figure 6, all permitted areas for physical parameters are absent for the case. Moreover, for all values of the ${\displaystyle \frac{L}{B}}$ ratio, an overproduction of techniparticles can be observed.

To estimate the dependence of the density ratios on the mass difference, we can introduce some useful parameterizations (5):

$$\begin{array}{cc}\hfill & {\sigma }_{N,E}={\sigma }_f\left({\displaystyle \frac{m}{T_f^{*}}}\right)+n,e,\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill & {\sigma }_{UU}={\sigma }_b\left({\displaystyle \frac{m}{T_f^{*}}}\right)+2u,\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill & {\sigma }_{UD}={\sigma }_b\left({\displaystyle \frac{m}{T_f^{*}}}\right)+u+d,\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill & {\sigma }_{DD}={\sigma }_b\left({\displaystyle \frac{m}{T_f^{*}}}\right)+2d.\hfill \end{array}$$

(26)

The reference value ${\displaystyle \frac{m}{T_f^{*}}}={\displaystyle \frac{1500}{250}}$ is assumed for the weight function.

The ratios of number densities as functions of these parameters are shown in Figure 7 and Figure 8 for $y=3$, ${\displaystyle \frac{L}{B}}=1$ and ${\displaystyle \frac{L}{B}}=0.8$, correspondingly. Red dotted lines depicted these ratios in the equal mass approximation, $m_N=m_E=m_{UU}=m_{UD}=m_{DD}$.

If, however, we analyze an opposite case of high mass difference for techniparticles, it results in an increase in the number density values. In some cases (as is seen, for example, in “d-u” plots), the sign of this density ratio can change. Consequently, the permitted region boundaries, depending on the density ratio values, can be shifted sufficiently.

As a limiting case, it is reasonable to consider the scenario of a baryon-symmetric primordial universe, (${\displaystyle \frac{L}{B}}=1$). Then, we have a complete absence of the technibaryon component suppression for an arbitrary relationship between techniparticles masses:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\displaystyle \frac{TL}{TB}}=3{\displaystyle \frac{{\sigma }_E}{{\sigma }_{UU}}}{\displaystyle \frac{y\left(\frac{L}{B}+3\right){\sigma }_{UU}+\frac{L}{B}-1}{3y\left(\frac{L}{B}+3\right){\sigma }_E-\frac{L}{B}+1}}=1.\end{array}\end{array}$$

(27)

Then, the ratio of densities should be written in the form:

$${\displaystyle \frac{{\Omega }_{DM}}{{\Omega }_b}}=6{\displaystyle \frac{m_{UU}+2m_{N/E}}{m_p}}{\displaystyle \frac{{\sigma }_{UU}{\sigma }_E}{{\sigma }_{UU}+3{\sigma }_E}}.$$

(28)

However, this scenario results in WIMP-like composite states, which present the hidden mass candidates, and here, we do not consider a more or less known model of WIMP dark matter.

In the WTC scenario, equations for $B,L$ parameters have the following form:

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill B-L& =4B+{\displaystyle \frac{({\sigma }_{UU}+3{\sigma }_E)y}{(y{\sigma }_{UU}+1){\sigma }_E}}TL-{\displaystyle \frac{4}{y{\sigma }_{UU}+1}}B\hfill \\ \hfill & =4B+{\displaystyle \frac{3({\sigma }_{UU}+3{\sigma }_E)y}{\sigma UU(3y{\sigma }_E-1)}}TB+{\displaystyle \frac{4}{3y{\sigma }_E-1}}B;\hfill \end{array}\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill B+L& =-2B-{\displaystyle \frac{({\sigma }_{UU}+3{\sigma }_E)y}{(y{\sigma }_{UU}+1){\sigma }_E}}TL+{\displaystyle \frac{4}{y{\sigma }_{UU}+1}}B,\hfill \\ \hfill & =-2B-{\displaystyle \frac{3({\sigma }_{UU}+3{\sigma }_E)y}{{\sigma }_{UU}(3y{\sigma }_y-1)}}TB-{\displaystyle \frac{4}{3y{\sigma }_E-1}}B,\hfill \end{array}\hfill \end{array}$$

(30)

where the coefficients are hyperbolically decreasing functions of the charge parameter. However, an increase in the mass of highly charged techniparticles can compensate for and correct this drop in the coefficients of the above equations.

Moving in the opposite direction and starting from the known density ratio, we can find other parameters for the scenario considered. Namely, when the technibaryonic component of the DM is strongly suppressed, i.e., when ${\displaystyle \frac{{\sigma }_{UU}}{{\sigma }_E}}\to 0$, for the ${\displaystyle \frac{L}{B}}$ ratio, we obtain:

$${\displaystyle \frac{L}{B}}={\displaystyle \frac{m_{N/E}(y{\sigma }_{UU}+1)}{\mp 5.043450334y-m_{N/E}(3y{\sigma }_{UU}-1)}}.$$

(31)

Here, the sign of $m_p$ is opposite to the sign of the technileptons excess charge, which can result from the sphaleron process. Therefore, we obtain the following value: ${\displaystyle \frac{{\Omega }_L}{{\Omega }_b}}={\displaystyle \frac{{\Omega }_{DM}}{{\Omega }_b}}={\displaystyle \frac{0.265}{0.0493}}$. Also, as a consequence of high technileptons mass, ${\displaystyle \frac{L}{B}}\to 1$.

2.2.2. Sphaleron Freezing out after the EWPT

The sphaleron rate strongly depends on the plasma temperature. And if the temperature is over the critical point—the EW phase transition temperature—the solution of the system of equations has changed and is the following:

$${\displaystyle \frac{TB}{B}}=-\alpha \left({\displaystyle \frac{L}{B}}+\gamma {\displaystyle \frac{TL}{B}}+\beta \right).$$

(32)

Now, the expressions for $\alpha ,\beta ,\gamma$ functions are cumbersome even in the most simple case of equal techniparticles masses:

$$\begin{array}{cc}\hfill & \alpha ={\displaystyle \frac{{\sigma }_{UU}}{3}}{\displaystyle \frac{({\sigma }_t+5)(2{\sigma }_{UU}+{\sigma }_E)+6({\sigma }_t+17)}{(9({\sigma }_t-1)y+2({\sigma }_t+5)){\sigma }_{UU}+({\sigma }_t+5){\sigma }_E+3(5{\sigma }_t+31)}},\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill & \beta ={\displaystyle \frac{18(2{\sigma }_{UU}+{\sigma }_E+18)}{({\sigma }_t+5)(2{\sigma }_{UU}+{\sigma }_E)+6({\sigma }_t+17)}},\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill & \gamma ={\displaystyle \frac{2({\sigma }_t+5){\sigma }_{UU}+(27(1-{\sigma }_t)y+{\sigma }_t+5){\sigma }_E+3(5{\sigma }_t+31)}{{\sigma }_E(({\sigma }_t+5)(2{\sigma }_{UU}+{\sigma }_E)+6({\sigma }_t+17))}}\hfill \end{array}$$

(35)

The reason is the nonzero mass of the $t-$ quark. Taking the limit $m_t\to 0$ or ${\sigma }_t\to 1$ (it results in the same limiting case), we find Equation (32), which coincides with the one from Ref. [13]:

$${\displaystyle \frac{TB}{B}}={\displaystyle \frac{{\sigma }_{UU}}{3}}\left({\displaystyle \frac{L}{B}}+{\displaystyle \frac{1}{{\sigma }_E}}{\displaystyle \frac{TL}{B}}+3\right).$$

(36)

In Figure 9, functions $\alpha$, $\beta$, $\gamma$ are depicted depending on the mass supposing $m_{UU}=m_{N/E}$. Despite the nonzero $t-$quark mass, the behavior of these coefficients is analogous to the above version (cf. with the Equation (36)). In particular, function $\beta$ has the asymptote $\beta ={\displaystyle \frac{54}{{\sigma }_t+17}}\approx 3$.

As follows from Equations (32) and (36), the density ratio depends on the charge precisely in the case of the nonzero mass of the $t-$ quark. However, this dependence is sufficiently weak (for all $\left|y\right|<100$), so this is insignificant for the quantitative details and qualitative inferences of the consideration.

Furthermore, in the case of high mass difference in comparison with the mass, Equation (32) is rewritten as the following:

$${\displaystyle \frac{{\Omega }_{DM}}{{\Omega }_b}}\approx -{\displaystyle \frac{3m_{N/E}}{m_p}}{\displaystyle \frac{1}{\gamma }}\left({\displaystyle \frac{L}{B}}+\beta \right).$$

(37)

Note also some special cases: when ${\displaystyle \frac{L}{B}}<-\beta \approx -3$, it is necessary to use only $y>0$ to prevent an anomalous isotope production; in the opposite situation, when ${\displaystyle \frac{L}{B}}>-\beta \approx -3$, it should be supposed that $y<0$.

Using the approximation $m_{N/E}={\displaystyle \frac{m_{UU}}{2}}$ from Equation (37), we obtain the density ratio dependency on the mass, which is plotted in Figure 10. Importantly, the ratio ${\displaystyle \frac{{\Omega }_{DM}}{{\Omega }_b}}$ can be made very close to its observed value; however, to realize this scenario, the necessary mass value should increase, along with the growth of the density ratio ${\displaystyle \frac{L}{B}}$. In Figure 10, it looks like a shift to the left of the intersection point of the line ${\displaystyle \frac{{\Omega }_{DM}}{{\Omega }_b}}=5.375$ and the graph of the function ${\displaystyle \frac{{\Omega }_{DM}}{{\Omega }_b}}$. Such analysis allows us to determine an upper mass limit. Thus, if ${\displaystyle \frac{L}{B}}<{10}^7$–${10}^8$ [28], the technileptonic X-helium mass should be $m_{N/E}<5$–$8\mathrm{TeV}$.

As above, the density ratio sign correlates with the sign of technilepton number for the technileptons excess. If the situation with this excess takes place, the sign can be determined by analyzing the ratio ${\displaystyle \frac{L}{B}}$ value.

Linear combinations of the baryon and lepton numbers densities, their sum and difference, have the most simple presentations in the case ${\sigma }_t\to 1$:

$$\begin{array}{cc}\hfill & B-L=4B+{\displaystyle \frac{3}{{\sigma }_{UU}}}TB+{\displaystyle \frac{1}{{\sigma }_E}}TL,\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill & B+L=-2B-{\displaystyle \frac{3}{{\sigma }_{UU}}}TB-{\displaystyle \frac{1}{{\sigma }_E}}TL.\hfill \end{array}$$

(39)

At the same time, these expressions differ significantly from Equations (29) and (30) which were used for case $T>T_c$.

2.2.3. Discussion

The above analysis of the sphaleron thermodynamics in the framework of the WTC scenario results in the value of the densities ratio, ${\displaystyle \frac{{\Omega }_{DM}}{{\Omega }_b}}$, known from astrophysical data. Taking the sphaleron freeze-out at a higher temperature than the EWPT, we find a few various types of (composite) techniparticle states constituting the DM. Also, an occurrence of the technileptonic X-helium is possible, and these dark atoms can be formed at any value of the charge parameter y; though, if $y<0$, additional WIMP-like states can emerge too.

This scenario, however, includes the dangerous process of DM overproduction, which can sufficiently constrain a set of values of the model parameters. We can extract an important prediction for the total mass lowest value: $m\ge 1\mathrm{TeV}$. Certainly, some areas of the parameter space are prohibited because they lead to high densities of anomalous isotopes. An upper limit for the masses of the DM candidates, unfortunately, can not be determined in a general case but only in some special cases.

Another important result is the very weak dependence of the density ratios on the charge parameter if the shpaleron freeze-out occurs after the EWPT.

In the general case, we can not establish any constraints for the masses of technileptons, techniquarks, and their bound states, as for the dark atoms masses, but when the density of technibaryon states is suppressed, i.e., ${\Omega }_{UU}\to 0$, we obtain some reasonable upper limit: $m<8\mathrm{TeV}$. Removing this density suppression leads to stronger mass restrictions for the prevention of DM overproduction; moreover, some unwanted production of heavy techniquark WIMP-like bound states is possible.

Also, an important conclusion is that the appearance of the charge dependence strongly follows from some additional requirements, for example, from the condition ${\mu }_0=0$ when $T>T_c$. Consequently, such extra conditions result in significant changes in the DM structure.

2.3. The Rate of Sphalerons

The possibility of processes occurring in cosmological plasma, as a result of which a certain connection is provided between changes in baryon and lepton numbers, is actually a consequence of the complex structure of the physical vacuum. In fact, a vacuum is a system of topologically nonequivalent interacted subsystems, which are characterized by integer values of the Chern–Simons numbers. Interactions of these vacuum states mean that, for the many-particle system (cosmological plasma), there is a possibility of transforming into a system with another vacuum state. Such transitions between different (more exactly, neighboring) vacuum states can significantly change a set of plasma quantum fields at its evolution in space and time.

The temperature of the cosmological plasma is a particularly important parameter, defining not only the restructuring of the system during phase transformations but also the rate of transition between different vacuum states. Moreover, the mechanism of such a transition is strongly different in various temperature ranges; in particular, at high temperatures, when the probability of macroscopic fluctuations in the system is not small, it is possible to determine the rate (i.e., normalized probability) of the so-called classical transitions (sphalerons) of the system into the neighboring vacuum with other topological characteristics. In this case, the system should pass through the maximum (the saddle point) of the potential. This process is the many-particle configuration emergence with certain quantum numbers and its subsequent decay into another, neighboring vacuum. The sphaleron solution of the field equations mean that the collective excitation of the quantum fields system with a certain set of quantum numbers (namely, baryon and lepton numbers) transforms the system of fields, providing the following conditions $\Delta (B-L)=0;\Delta (B+L)\ne 0$.

Therefore, these classical sphaleron trajectories passing through the saddle point of the scalar potential transform the multi-particle state at high temperatures into another state with different baryon and lepton numbers. Together with the obligatory fulfillment of Sakharov conditions, these vacuum–vacuum transitions result in the necessary interdependence of baryo- and leptogenesis, and consequently, they can provide the correct value of the baryon asymmetry (BAU) of the Universe. Of course, the character and rate of this many-particle process are mainly determined by the shape of the scalar potential at various scales.

Note, however, at temperatures close to zero, the transition between neighboring vacuum states is based on the quantum mechanism of tunneling transitions under a potential barrier; these are the so-called instanton transitions, i.e., well-known classical solutions of nonlinear equations, localized in space and in the imaginary, i.e., classical (Euclidean) time.

Probabilities of instantons, i.e., tunneling transitions at $T=0$, were calculated and discussed in a lot of papers, for example, in Refs. [29,30] and in the important review [17]. As for sphalerons, their rate at nonzero, T, can be analyzed using a thermodynamic approach, as a probability per unit time and volume of high-energy multi-particle fluctuation. The first important results on the dependence of the sphaleron rate on temperature, the form of the potential and the dynamics of quantum fields are summarized in detail in Ref. [17] (see also references therein); these quantitative results were derived in the SM or some of its extensions. Here we only consider the WTC scenario.

Therefore, we should work in the following sequence: introduce an extension of the SM with some set of additional degrees of freedom (to obtain the possible DM candidates), determine the corresponding scale of energies for this scenario; and consequently, determine the range of temperatures at which these new states would contribute significantly to the dynamics of cosmological plasma between and near the phase transition points. Most significant for the next consideration are the local and global characteristics of the potential, i.e., the structure of a vacuum and scalar sector of the model.

Then, the rate of sphaleron appearance and the decay in the dependence of temperature can be estimated using various approaches (for example, in the framework of nonperturbative lattice models or from the Schrodinger equation with potential modeling of the periodicity of the physical vacuum structure). Furthermore, the dynamics of the SM scalar sector can be enriched by additional Higgs doublets, the effects of a triple Higgs interaction, or the appearance of a new scalar field in the period between the inflation and radiation epochs [31]. Efforts (and hopes) to amplify the EW baryogenesis magnitude by intensive sphaleron transitions also rely on an extension of the set of fermions in a scenario: transition from the SM to the fourth generation of heavy quarks or to techniquarks and composite scalars. Correspondingly, a new scale of energies and new types of bound states arise.

Note, to provide a cosmological plasma disequilibrium, as one of the obligatory conditions by Sakharov, is also possible in the sphaleron decoupling regime [32,33]. This means that sphaleron transitions can turn off gradually, starting from the temperature of the EW phase transition up to their freeze-out point. Obviously, such a decoupling process depends on the sphaleron parameters, in particular, their rates and sizes.

At high plasma temperatures, $T\ge E_{sph}$ (sphaleron energy is estimated as the height of the potential barrier), in the framework of quasi-classical methods [17,34,35], the width of sphaleron decay can be found in the following form:

$$\Gamma \sim {\left({\displaystyle \frac{m_W^2}{{\alpha }_{W}T}}\right)}^2·exp(-E_{sph}/T).$$

Of course, given the classical (sphaleron) solutions for the field equations, we can integrate over zero modes to obtain the corresponding normalizing factors. Furthermore, this expression for ${\Gamma }_{sph}$ does not allow us to analyze the possible contributions of higher loops.

The sphaleron rate can be considered analytically, based on effective theory [14,36,37,38], when the BAU arises in the hot phase of the plasma. The generality of this approach means that the considered scheme of calculations can be formulated for the $SU\left(N_C\right)$ group, so the sphaleron rate will depend on the $N_C$.

The most direct method of estimating the sphaleron rates is the Green functions application; this is because the Yukawa coupling for the Higgs field is connected with the rate of sphaleron transition, which describes the baryon number violation (see [17] and references within). The proportionality of these quantities follows from the calculation of the classical correlation function.

Then, it seems natural that the classical process of transition via the saddle point of the model potential realizes a relationship between the rate of fermion number violation and the classical correlation function. At high temperatures, ($gT>>m_W\left(T\right)$), it provides the sphaleron decay width

$${\Gamma }_{sph}={\left({\alpha }_{W}T\right)}^{4}F({\displaystyle \frac{\lambda }{g^2}},{\displaystyle \frac{m^2\left(T\right)}{g^4{T}^2}}),$$

and in the limit of very high temperatures, $F\to 1$. It should be added that this form of ${\Gamma }_{sph}$ can be derived from dimensional considerations, given that in the restored symmetry phase, there is no exponential suppression.

An effective new scale ∼$gT$ was found to work in the Hamiltonian approach [39,40], and the reason is obvious—in this approach, thermal loops arise that contribute to the sum of zero modes with a thermal weight; however, these terms come with some ultraviolet divergences. In this way, a cutoff scale is introduced to regularize a hard momenta region. As a result, a method can be formulated for the real-time calculations of baryon number violation processes at high temperatures considering classical equations of motion, with an account of the initial conditions. In other words, this scheme presents a reasonable approach to quantitatively analyzing the infrared behavior of real-time correlation functions.

Thus, in the EW-restored symmetry phase, the rate of sphaleron transitions gives the following functional form: $\Gamma =k·{\left({\alpha }_{W}T\right)}^4$ with factor $k\approx$ 0.1–1.0. Nevertheless, the question of the magnitude of the spontaneous CP-breaking quantity remains, as well as some important questions about the feasibility of Sakharov’s conditions (for example, the required degree of cosmological plasma instability and disequilibrium at high temperatures).

As was mentioned above, the form of the scalar potential model strongly determines the sphaleron parameters. Studying the minimal composite Higgs model [41], a second sphaleron solution with high energy, $E_{sp{h}_2}\approx$ (3–8)$E_{sp{h}_1}$, and a half-integer Chern-Simons number was found. Moreover, the energy of this additional sphaleron is linearly proportional to the compositeness scale. The global structure of the scalar potential can also result in sphalerons of energy that are lower than in the SM—in fact, all of the new types of sphaleron transitions with the extended energy spectrum correspond to possible deformations or modifications of $V\left(\varphi \right)$ due to an extra scalar degree of freedom [42].

The emergence of a few sphaleron types at various scales can be understood as a consequence of the complex nonperturbative structure of vacuum. Its characteristics, v.e.v’s like (techni-)quark and (techni-)gluon condensates, in reality, should be non-local and non-static dynamical functions. Then, the sphaleron branches of different energies correspond to classical solutions at temperatures $T\gtrsim m_{sph}\sim <\overline{Q}Q>$, where the operator $<\overline{Q}Q>$ characterizes the Higgs vacuum averages in the compositeness scenario. For the technicolor model, a reasonable estimation [43] is: ${\Lambda }_{TC}\sim$ 400–500 GeV, $<\overline{Q}Q>\sim {\Lambda }_{TC}^3,v=246\mathrm{GeV}$.

The classical nature of sphaleron solutions supposes an effectiveness of the lattice approach for the calculation of sphaleron rates. A review of the lattice results was done in Ref. [17]; some of the first quantitative data for vacuum–vacuum zero-mode contributions into pre-factors were provided for the rates depending on the temperature in the symmetric and broken phases.

Studying the pure-glue strong $SU\left(2\right)$ scenario, lattice methods for calculating sphaleron rates at different temperatures in symmetric and broken phases were sufficiently developed and improved [24,44,45,46]. To date, the lattice methods have been able to reasonably predict the temperature dependence of the scalar field condensate and results of non-perturbative summation of small fluctuations, i.e., pre-factors for the sphaleron rate.

For the scalar field vacuum average, $v\left(T\right)$, the temperature dependence had been calculated at the lattice for the electroweak crossover at $T_c\approx 159\mathrm{GeV}$; then, in the symmetric phase ${\Gamma }_{sph}/T^4\approx (18\pm 3){\alpha }_W^5$ and is almost constant, in addition, in the broken phase, $140\mathrm{GeV}\le T\le 155\mathrm{GeV}$ a stable expression was found: $ln\Gamma /T^4=(0.83\pm 0.01)T-(147\pm 1.9)$ [24]. Using a lattice approach, an estimation of the freeze-out temperature in the early Universe was found (i.e., the cosmological plasma temperature at which the Hubble rate exceeds the rate of the baryon number violation): $T_f=(131.7\pm 2.3)\mathrm{GeV}$.

In the pure gauge case at $T\approx 1.24T_c$, where $T_C\approx 300\mathrm{MeV}$, for QCD sphalerons, the lattice calculations in the symmetric phase are [45]: ${\Gamma }_{sph}=0.079\left(25\right)·T^4.$ Sphaleron rates for the case of (2 + 1) QCD were also calculated at different temperatures using the lattice methods [46].

For suggestions for sphaleron observation experiments, it is necessary to study sphaleron production cross-sections as a function of sphaleron energy, temperature, and phases in various scenarios. Thus, by comparing lattice calculations of sphaleron parameters, and it was found that the results of different groups were in reasonable agreement for the same phases and regions of temperatures. With such a calculational basis, it is possible to formulate ideas about searching for sphaleron in high-energy experiments.

A magnetic field that is sufficiently strong is an important component of the dynamics of the hot Universe. It is clear that the magnetic field should affect the cosmological plasma fluctuations, especially in the neighboring phase transition points, introducing correlations into the interactions of the charged components of the plasma and their motion. If strong local fluctuations of plasma energy density occur at high temperatures, the sphaleron transitions can be generated, and due to an emerging imbalance in left- and right-handed quarks densities, an electric current, which is parallel to the plasma magnetic field, is produced. Thus, a so-called chiral magnetic effect also occurs, and the electromagnetic characteristics of the hot cosmological plasma are changed; moreover, local changes of $B,L$ quantum numbers are possible as a consequence of the emergence of sphalerons. The rate of baryon number violation due to the effect of an external magnetic field [32,47] for temperatures in the vicinity of the crossover, mostly in the broken phase, can also be derived using the lattice approach as a function of temperature and the magnetic field strength [48]. Namely, the critical temperature should decrease with an increase in an external magnetic field.

Thus, resulting from the modifications of the scalar potential, the energy of a sphaleron can be drastically changed—it can decrease for some types of the standard $V\left(\varphi \right)$ deformations [42] or strongly depend on the ternary coupling of the scalar interaction coupling, $\Delta E_{sph}\sim -{\lambda }_3$, in the scenario with an extended Higgs sector [49,50]. This means that information on the scalar potential form can be derived from data about the energy of sphaleron transitions.

To extract these data from collider experiments or astrophysical observations, there are numerous ideas and suggestions [15,16,51,52,53,54]. Ground-based installations and measuring complexes, such as LHC, IceCube, LHAASO and others, generate and collect a huge amount of information about high-energy multi-particle reactions, neutrinos, and photon scattering, and space telescopes can also select specific signals induced by cosmic rays of high energies. These data can be used for the search and interpretation of multi-particle events, which are characteristic of the sphaleron processes based on their calculated rates or cross-sections.

Refs. [53,55,56] proposed a new interesting approach that considers the physical vacuum as a set of subsystems so that the potential is a periodic structure along the axis of Chern–Simons vacuum numbers. This form of the potential results in the Schroedinger equation for the wave function of Bloch type, following from Bloch theorem for the solid-state periodic lattice. The solution of the wave equation then allows for calculating the probability of transition between neighboring vacua states, i.e., the sphaleron rate. Therefore, this description of a complex vacuum structure is obviously optimistic as a basis for an experimental search for sphalerons.

The solution, however, leads to an unexpected claim: the sphaleron transition is unsuppressed in the broken phase for energies above $E_{sph}$. This contradicts the earlier-predicted exponential damping in this region, which seems reasonable. However, this is not the final problem of the model: the resonance amplification of the transition rate between vacuum subsystems was reanalyzed [57], and it was found that some time-dependence was missed in this approach; furthermore, a detailed study of transitions between coherent states in WKB approximation shows an exponential suppression of the sphaleron processes. Therefore, we again come to a more pessimistic claim about the probability of sphaleron observation in high-energy processes. In fact, the situation remains undefined, and the task of reliably calculating of sphaleron rate and cross section is not fully solved.

Hot cosmological plasma involves a strongly interacting QCD (techni-QCD) subsystem; consequently, strong sphaleron transitions could occur there at appropriate temperatures. There is a well-known axion (or axion-like) scenario, with these super-light objects as the DM candidates. It was discovered [58] that QCD sphalerons at high energies should significantly affect the axion processes—the axion density and, correspondingly, reactions involving them significantly depend on the rate of strong sphalerons.

Questions arise regarding extended SM scenarios with additional (heavy) fermions (techniquarks, for instance)—could extra fermions with non-standard interactions impact the vacuum–vacuum transition rate, and if so, how? Could the shortest path above the potential barrier between neighboring topologically distinct vacua be deformed by additional fermion fields [59,60,61]? Some results seem important for the next analysis: if an extra heavy fermion doublet with $m_F\sim \mathrm{TeV}$ is added, the height of the potential barrier could decrease significantly with the growth of the fermion mass; moreover, if $m_F\sim 10\mathrm{TeV}$ an additional sphaleron branch could occur due to potential barrier deformations. Furthermore, at the saddle point of the barrier, the sphaleron multi-particle system no longer has reflection symmetry, and the Chern–Simons number for the sphaleron solution is $n_{CS}\ne 1/2$. This also results from interaction with fermions. It was also noted [59,60] that in the EW theory, the fermion sea noticeably impacts the form of the potential barrier and sphaleron rate at non-zero temperatures, which leads to a suppression of the transition rate.

3. Dark Atom Formation at BBN

3.1. Bohr-like OHe and Thomson-like XHe Atoms

A system of dark atoms is comprised of $-2n$ (n is any natural number) charged X particles (for $n=1$, corresponding to $O^{--}$) and n nuclei of ${}^4\mathrm{He}$, held together by the Coulomb force. As per the X-helium framework, a charged stable particle of X can manifest either lepton-like characteristics or act as a unique aggregation of new families of heavy quarks characterized by reduced interactions with hadrons [7]. Empirical observations indicate that the minimal mass of multiply charged stable X particles approximates $1\mathrm{TeV}$ [1]. Additionally, within the masses of these techniparticles, their cumulative mass significantly contributes to the overall density of non-relativistic matter, aligning with the observed characteristics of dark matter. Given that the mass of these heavy $-2n$ charged particles dictates the mass of dark atoms, they consequently elucidate the observed density of all dark matter.

The specific configuration of a bound system of a dark atom is dictated by the parameter $a\approx Z_{\alpha }{Z}_X\alpha A_{\alpha }{m}_p{R}_{nHe}$, where $\alpha$ denotes the fine structure constant, $Z_X$ and $Z_{\alpha }$ represent the charge numbers of particle X and the $nHe$ nucleus, respectively, $m_p$ signifies the proton’s mass, $A_{\alpha }$ signifies the mass number of the $nHe$ nucleus, and $R_{nHe}$ denotes the radius of the corresponding $nHe$ nucleus. The physical significance of the parameter a (not $\alpha$) is the ratio of the Bohr radius of the dark atom to the radius of the n-helium nucleus. For instance, in the case of OHe, this parameter elucidates the proportion of the Bohr orbit for the $\alpha$-particle within a dark atom to the radius of the helium nucleus. Should the Bohr orbit of XHe prove smaller than the dimension of the n-helium nucleus, then the dark atom assumes the structure of a Thomson atom.

When parameter a falls within the interval $0<a<1$, the resulting bound state exhibits a structure reminiscent of a Bohr atom, featuring a centrally located negatively charged X particle and a point-like helium nucleus orbiting around it akin to Bohr’s atomic model. Conversely, for values of a within the range $1<a<\infty$, the bound states take on characteristics akin to Thomson’s model of atoms, wherein the non-point $He$ nucleus undergoes vibrations around a substantially massive negatively charged X particle.

The characteristics inherent in X-helium atoms, encompassing both Thomson and Bohr models, effectively impede the cosmic dark atom flow, decelerating it to thermal energies and ensuring its slow diffusion towards the Earth’s core. This deceleration process complicates the direct detection of dark matter particles utilizing techniques based on recoil effects resulting from WIMP–nucleus interactions [62].

The OHe configuration comprise two constituents: the helium nucleus and the $O^{--}$ particle, both of which are mutually bound and regarded as point particles. Within this system, a spherical coordinate system is established with its origin situated at the $O^{--}$ particle. In this way, in a scenario where $Z_{\alpha }=2$ and $Z_X=-2$, the model considers the $He$ nucleus as a point-like entity randomly traversing along the surface of a sphere with a radius equal to the Bohr atom radius $R_b$. Consequently, the binding energy for OHe, incorporating the point-like charge of ${}^4\mathrm{He}$, amounts to $I_0\approx 1.6\mathrm{MeV}$, while the Bohr radius for $He$ is approximately $R_b\approx 2\times {10}^{-13}\mathrm{cm}$ [7]. The helium nucleus sustains a constant speed called the Bohr velocity, denoted as $V_{\alpha }$. The velocity of the $\alpha$-particle moving along the Bohr orbit corresponds to approximately $V_{\alpha }\approx 3\times {10}^4\mathrm{cm}/\mathrm{s}$ [63].

The instance of $-2$ charged particles represents just one particular case. In the area of the “new” physics we are examining, the particle of X can carry a charge of $-2n$, thereby forming X-helium dark atoms with n nuclei of ${}^4\mathrm{He}$. These X-helium dark atoms themselves, commencing from $n=2$, embody Thomson atoms. The X-helium dark atoms in Thomson’s model comprises two bound constituents: the n-helium nucleus and the X particle. In this configuration, XHe is structured as follows: the charged n-helium nucleus, representing a charged sphere, encases the point-like particle X within it. The $nHe$ is notably lighter than X, resulting in the nuclear cluster oscillating around X. In ref. [63], which describes a system of three interacting charged particles through Coulomb and nuclear forces, a numerical model is sequentially constructed that can qualitatively reconstruct the shape of the effective interaction potential between XHe and the target nucleus in two semiclassical approaches: the approach of reconstructing particle trajectories, which includes the Bohr model and the Thomson model, and the approach of reconstructing the potential.

3.2. Formation of Bound States

Heavy negatively charged particles $X^{-2n}$ should capture a primordial light nucleus at the nucleosynthesis stage to form neutral bound states. The energy and momentum transfer between such dark atoms $XHe$ with mass $m_{XHe}$ and baryons in primordial plasma is effective [7] only when

$$n_B〈\sigma v〉(m_p/m_{XHe})t>1.$$

(40)

When this condition violates at temperature $T_{od}\approx S_3^{2/3}\mathrm{keV}$, $S_3=m_{XHe}/\left(1\mathrm{TeV}\right)$, dark atom gas decouples from the plasma. The total cross section could be estimated as

$$\sigma \approx {\sigma }_o\sim \pi R_{XHe}^2\approx {10}^{-25}{\mathrm{cm}}^2=250{\mathrm{GeV}}^{-2}$$

(41)

and generally depends on the charge of the heavy core. Indeed, if the nuclear shell is symmetric in the number of protons and neutrons, then $R_{XHe}\approx R_{nHe}=r_0{A}^{1/3}=r_0{\left(4n\right)}^{1/3}$. However, the considered bound states could still interact with the plasma while

$$n_{XHe}\sigma vt>1,$$

(42)

where the dark atom concentration and the interaction time could be found as

$$n_{XHe}={\displaystyle \frac{{\rho }_c}{m_{XHe}}}{\Omega }_{DM}^{\mathrm{now}}{\left({\displaystyle \frac{T}{T_{\mathrm{now}}}}\right)}^3,$$

(43)

$$t={\displaystyle \frac{3}{4}}{\displaystyle \frac{m_{Pl}}{T^2}}\sqrt{{\displaystyle \frac{5}{{\pi }^3{g}_{*}}}}.$$

(44)

Here, $m_{Pl}$ is a Planck mass and $g_{*}$ is a number of ultrarelativistic degrees of freedom at the considered temperature [64]. The average velocity of a non-relativistic light nucleus with mass $m_N$ is just $<v>=\sqrt{{\displaystyle \frac{8T}{\pi m_{\mathrm{N}}}}}$.

Therefore, some new processes should take place at the temperature $T\sim 100keV$ if the following condition is fulfilled:

$$n_{XHe}\sigma vt={\displaystyle \frac{3}{2{\pi }^2}}{\displaystyle \frac{{\rho }_c{\Omega }_{DM}^{\mathrm{now}}{m}_{Pl}\sigma }{m_{XHe}{T}_{\mathrm{now}}^3}}\sqrt{{\displaystyle \frac{10T^3}{g{m}_{\mathrm{N}}}}}\approx 3040.3{\displaystyle \frac{\sigma }{m_{XHe}\sqrt{m_{\mathrm{N}}}}}>1.$$

(45)

This condition is satisfied not only for protons and helium but also for $XHe$ with mass $m_{XHe}<$ 10–20 TeV. It was shown earlier that the density of dark matter formed during sphaleron transitions is either too low or too high if the new particles X are too heavy. Therefore, the scenario in which the merging of dark atoms occurs in the early universe seems probable.

In addition to the standard nucleosynthesis reactions, the dark atom model predicts the following processes:

$$\begin{array}{cc}\hfill & X+N\to XN+\gamma ,\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill & X{N}^1+N^2\to X{N}^3+\gamma \setminus N^4,\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill & X{N}^1+X{N}^2\to X_2{N}^3+\gamma \setminus N^4.\hfill \end{array}$$

(48)

The first two reactions describe the dark atom formation. Unfortunately, reaction (47) could result in the overproduction of anomalous isotopes and primordial metals. Such a possibility has been considered for doubly charged cores $O^{--}$ in a recent paper [65]. However, in the case of large nuclear charges, the particular set of reactions may be different. First of all, the neutral (or positively charged) state should form at lower temperatures since the number of required reactions has increased. It can also be expected that the primordial metals overproduction could be suppressed due to an increase in the binding energy. Reaction (48) describes the formation of a molecule-like structure. The number of merged dark atoms can not be high due to a violation of condition (45).

To find out how dark atoms affect the nucleosynthesis, it is necessary to solve the following system of kinetic equations:

$${\displaystyle \frac{d{n}_i}{dt}}+3H{n}_i=\sum _{j,k}{n}_j{n}_k{\left(\sigma v\right)}_i^{jk}-n_i\sum _j{n}_j{\left(\sigma v\right)}^{ij},$$

(49)

where $n_i$ is the density of the $i-$type particles. The second term on the left side of each equation describes the density change due to the expansion of the Universe. On the right sides of these equations, all processes involving particles of $i-$type are taken into account. Certainly, the list of all possible reactions, as well as the considered temperature range, depends on the structure of the dark atom. For instance, processes of the type (46) become thermodynamically advantageous at different temperatures determined by the binding energy. This means that the capturing of protons should start much later than the primordial helium capturing. Moreover, it is necessary to take into account the suppression of Coulomb repulsion in a nuclear shell when analyzing the reactions of (47) and (48) types. Indeed, such screening should prevent the $\alpha -$decay of nuclei $X^{8}Be$. Then, for the semi-empirical estimation of the binding energy difference, we obtain:

$$\Delta W_N\lesssim 0.7Z^2{A}^{-{\displaystyle \frac{1}{3}}}.$$

(50)

Here, we neglect the contributions of weak decays in the nuclei shell.

Finally, the system of Equations (49) should take into account non-equilibrium states called $N^4$ in (47) and (48). To describe these processes as the sources of such particles, it is necessary to introduce the momentum distributions

$${\displaystyle \frac{d{n}_N^{*}}{d{p}_N}}={\varphi }_N(p_N,t).$$

(51)

As a matter of fact, these functions of momenta are solutions of additional equations:

$${\displaystyle \frac{𝜕{\varphi }_{N_i}}{𝜕t}}=\sum _{j,k}{n}_j{n}_k{\displaystyle \frac{d{\left(\sigma v\right)}_{N_i}^{jk}}{d{p}_{N_i}}}-{\varphi }_{N_i}\sum _j{n}_j{\left(\sigma v\right)}^{N_{i}j}\left(p_{N_i}\right)-{\varphi }_{N_i}\sum _j\int {\varphi }_{N_j}{\left(\sigma v\right)}^{N_i{N}_j}d{p}_{N_j}.$$

(52)

The terms on the right side describe the non-equilibrium light nuclei produced in reactions (47) and (48), the interaction of these nuclei with equilibrium particles, and the scattering of each other, correspondingly. Therefore, the total concentration of N-type light nuclei can be written as the following sum: $n_N=n_N^{\mathrm{eq}}+n_N^{*}$.

The type of dark atom structure determines not only the energy release but also the cross sections. To find it, one should calculate the ratio of the nuclear and Bohr radii $a=Z_X{Z}_N\alpha m_N{r}_0{A}_N^{1/3}$. Table 2 shows the structure of some $XN$ bound states:

• B—all isotopes form Bohr-like particles;
• T—all isotopes form Thomson-like particles;
• Number—the mass number of the lightest Thomson-like particle.

All neutral and positively charged states at $n\ge 2$ should have a Thomson structure, but hydrogen mostly forms Bohr-type dark ions. This is consistent with the result of solving the two-body Coulomb problem obtained in [65].

Table 2. Structure of the $XN$ bound states.

n	A
	$\mathit{H}$	$\mathit{He}$	$\mathit{Li}$	$\mathit{Be}$	$\mathit{B}$	$\mathit{C}$
1	B	4	T	T	T	T
2	4	3	T	T	T	T
3	3	T	T	T	T	T
4	3	T	T	T	T	T
5	2	T	T	T	T	T

Table 2. Structure of the $XN$ bound states.

n	A
	$\mathit{H}$	$\mathit{He}$	$\mathit{Li}$	$\mathit{Be}$	$\mathit{B}$	$\mathit{C}$
1	B	4	T	T	T	T
2	4	3	T	T	T	T
3	3	T	T	T	T	T
4	3	T	T	T	T	T
5	2	T	T	T	T	T

The processes in (46) are similar to the standard recombination, especially for hydrogen and helium. As a result, one can make an estimation of the temperature $T_{\mathrm{rec}}$ at which the bound state formation becomes possible. Using equations

$$n_i^{\mathrm{now}}{\left({\displaystyle \frac{T}{T_{\mathrm{now}}}}\right)}^3=g_i\left({\displaystyle \frac{m_i{T}^{\frac{3}{2}}}{2\pi }}\right)e^{-{\displaystyle \frac{m_i}{T}}}$$

(53)

for non-relativistic equilibrium concentrations, the Saha equations [64] can be written out. Their solutions are

$$T_{\mathrm{rec}}=W_{X-N}{\left(ln\left({\displaystyle \frac{g_X{g}_N}{g_{XN}}}{\left({\displaystyle \frac{m_N{T}_{\mathrm{now}}^2}{2\pi W_{X-N}}}\right)}^{{\displaystyle \frac{3}{2}}}{\displaystyle \frac{1}{n_N^{\mathrm{now}}}}\right)\right)}^{-1}.$$

(54)

For the dark atom, $W_{X-N}$, binding energy for the system ”heavy core plus nuclear shell” depends on the type of this state and the number of spin degrees of freedom, $g_X$. The coefficient $g_{XN}$ is determined by a specific model of the dark atom. According to Table 2, the binding energy for half of the considered cases can be extracted from the simplified Bohr model

$$W_{X-N}^{\mathrm{Bohr}}=2n^2{Z}^2{\alpha }^2{m}_N={\displaystyle \frac{1}{2m_N{R}_b^2}}.$$

(55)

However, for the first simple estimation of the binding energy of Thomson-like bound states $W_{X-N}^{\mathrm{Thomson}}$, a mechanical model can be used. Assuming that a heavy core $X^{-2n}$ oscillates inside a nucleus N, the following Hamiltonian could be used

$$\mathcal{H}=\left\{\begin{array}{c}{\displaystyle \frac{P^2}{2m_N}}-{\displaystyle \frac{3nZ\alpha }{R_N}}+{\displaystyle \frac{nZ\alpha }{R_N}}{\left({\displaystyle \frac{r}{R_N}}\right)}^2,r<R_N\hfill \\ {\displaystyle \frac{P^2}{2m_N}}-{\displaystyle \frac{2nZ\alpha }{R_N}},r>R_N\hfill \end{array}\right.$$

(56)

where the radius of a light nuclei $R_N=r_0{A}^{{\displaystyle \frac{1}{3}}}$, P is momentum. Therefore, the classical estimation could be found

$$<U_{Coulomb}>=〈{\displaystyle \frac{2nZ\alpha }{2R_N}}\left(3-{\displaystyle \frac{R_N^2{sin}^2\left(\omega t\right)}{R_N^2}}\right)〉={\displaystyle \frac{5}{2}}{\displaystyle \frac{nZ\alpha }{R_N}},$$

(57)

However, there is another estimation that is usually used in the literature [66,67]. Following [66]

$$W_{X-N}^{\mathrm{Thomson}-\mathrm{Glashow}}=3{\displaystyle \frac{nZ\alpha }{R_N}}\left(1-\sqrt{{\displaystyle \frac{1}{2nZ\alpha m_N{R}_N}}}\right)=3{\displaystyle \frac{nZ\alpha }{R_N}}\left(1-\sqrt{{\displaystyle \frac{R_b}{R_N}}}\right).$$

(58)

Both these equations predict that, for the very light nuclei, the binding energy value of a Thomson-like atom is higher than for a Bohr-like atom. For heavier nuclei (like $Be$), the inverse ratio is correct. It could also be highlighted that this equation gives a bad prediction for boundary case $a={\displaystyle \frac{R_N}{R_b}}\approx 1$. Unfortunately, the obtained values of energy differ significantly from those obtained in [65]. A correct quantum description of the Thomson-like bound states is necessary to find a list of possible reactions and should be created in the future.

The calculation results are presented in Table 3. For Thomson-like dark atoms, both Equations (57) and (58) were used. The predicted temperatures are written out for $<U_{Coulomb}>/W_{X-N}^{\mathrm{Thomson}--\mathrm{Glashow}}$ correspondingly. The qualitative difference is present only for boundary cases with $a\approx 1.0-1.1$ (deuterium at $n=5$ and helium at small values of n) when the Bohr formula could also be used. The result also depends weakly on the values of $g_X$ and $g_{XN}$. Since helium-4 should be formed at temperature $T\approx 65\mathrm{keV}$ [64], there are two possible scenarios:

• For the $n<4$ nucleus, the ${}^4\mathit{He}$ capturing happens earlier than protons and/or deuterium can be captured. Then, the excess of dark ions, ${\left(XHe\right)}^{-2n+2}$, is formed to start reactions (47).
• For $n\ge 4$, hydrogen capturing becomes possible before the ${}^4\mathit{He}$ formation. Therefore, another branch of reactions ($XH+N\to \dots$) is started. However, processes involving hydrogen lead to an additional danger of overproduction of anomalous isotopes at later stages of the dark atom formation. This may help to set limits on the maximum charge of a heavy core.

In the calculations, the mass reduction for the helium nucleus was not taken into account. However, a simple estimation based on (50) demonstrates no significant changes.

There are a lot of various ways to form dark atoms in the early Universe. The type of process producing the dark atoms is determined mainly by the charge of the heavy core. However, to describe the following reactions of the formation of bound states, it is necessary to know the concentrations of primordial particles as a function of temperature. These dependencies can be obtained by solving the system of kinetic equations. Precise determination of the energies and cross sections of these reactions may require numerical simulations of light nuclei. It should be mentioned that problems of the bound-state formation in a strong Coulomb field also arise in QCD [68,69]. In the deconfinement phase, quarks interact only due to electromagnetic forces. The case of dark atoms is significantly distinguished by the absence of a known Lagrangian of the core–shell–nucleus system interaction. The introduction of form factors implies conducting a number of experiments, which is difficult to implement in the context of the structure of dark matter particles.

Table 3. The recombination temperatures for dark atoms.

n	${\mathit{T}}_{\mathbf{rec}}\mathbf{keV}$
	$\mathit{p}$	$\mathit{D}$	${}^3\mathit{He}$	${}^4\mathit{He}$
1	3	5	28	107/6
2	13	19	∼192/42	222/85
3	29	44	∼292/116	339/180
4	54	79	∼395/198	459/285
5	86	290/17	∼500/286	580/395

Table 3. The recombination temperatures for dark atoms.

n	${\mathit{T}}_{\mathbf{rec}}\mathbf{keV}$
	$\mathit{p}$	$\mathit{D}$	${}^3\mathit{He}$	${}^4\mathit{He}$
1	3	5	28	107/6
2	13	19	∼192/42	222/85
3	29	44	∼292/116	339/180
4	54	79	∼395/198	459/285
5	86	290/17	∼500/286	580/395

3.3. Dark Atom Scenario of Dark Matter

After their formation, dark atoms should be in thermal equilibrium with primordial plasma. At that time, as in the case of ordinary baryonic matter, the fluctuations of dark matter density are transformed by radiation pressure in acoustic waves within the cosmological horizon. However, dark atom gas decouples from plasma at a relatively high temperature $T_{od}\approx S_3^{2/3}\mathrm{keV}$. Therefore, it is involved only in the large-scale structure formation but not in the formation of molecular clouds, stars, planets and other astrophysical objects. The development of gravitational instability starts at the transition from radiation- to matter-dominated stages at the temperature $T\le T_{RM}\approx 1\mathrm{eV}$ (which is equivalent to $t\sim {10}^{12}\mathrm{s}$).

The need for bounding into a neutral state results in a specific dark atom scenario of dark matter. Due to the existence of an additional temperature scale $T_{XHe}$ corresponding to the dark atom formation, the small-scale structure is slightly suppressed. Indeed, on the one hand, at high temperatures $T>T_{RM}$ within the cosmological horizon $l_h=t$, the total mass of dark atoms is

$$M={\displaystyle \frac{4\pi }{3}}{\rho }_d{t}^3={\displaystyle \frac{4\pi }{3}}{\displaystyle \frac{T_{RM}}{T}}{m}_{Pl}{\left({\displaystyle \frac{m_{Pl}}{T}}\right)}^2,$$

(59)

where the density ${\rho }_d=(T_{RM}/T){\rho }_{tot}$. Provided that the decoupling temperature depends on the mass of dark matter particle significantly $T_{od}\approx S_3^{2/3}={\left({\displaystyle \frac{m_{XHe}}{1\mathrm{TeV}}}\right)}^{2/3}\mathrm{keV}$, it could be estimated

$$M_{od}={\displaystyle \frac{T_{RM}}{T_{od}}}{m}_{Pl}{\left({\displaystyle \frac{m_{Pl}}{T_{od}}}\right)}^2\approx 5\times {10}^{40}{S}_3^{-2}\mathrm{g}\approx {10}^7{S}_3^{-2}{M}_{\odot },$$

(60)

where the solar mass $M_{\odot }$ was extracted. This value should be rescaled for the temperature of dark atom formation $T_{XHe}\sim 50\mathrm{keV}$. Thus, $M_{XHe}=M_{od}{(T_{od}/T_{XHe})}^3\approx 3\times {10}^{37}\mathrm{g}$.

On the other hand, the propagation of sound waves with a relativistic equation of state $p=ϵ/3$ determines the maximal value of the Jeans length of a dark atom before decoupling ${\lambda }_J=l_h/\sqrt{3}=t/\sqrt{3}$. It is of the order of the cosmological horizon. Nevertheless, at $T=T_{od}$, the Jeans length should be much smaller ${\lambda }_J\sim v_{XHe}t,$ where the thermal velocity is $v_{XHe}=\sqrt{2T_{od}/m_{XHe}}.$ As a result, the Jeans mass in the dark atom gas decreases after decoupling

$$M_J\sim v_{XHe}^3{M}_{od}\sim 5\times {10}^{-13}{M}_{od}.$$

(61)

Therefore, it is a strong suppression of fluctuations with mass $M<M_{XHe}$. Moreover, there are almost no fluctuations on scales $M_{XHe}<M<M_{od}$ due to adiabatic damping of sound waves in plasma at the RD stage. As a result, the small scale structure formation could be suppressed. However, the temperature $T_{XHe}$ can only be well-defined for O-helium using the Saha equation (see Table 3). This means that not only are the detailed numerical simulations of the structure formation needed but also the description of modified nucleosynthesis. In any case, the free streaming suppression in ordinary Warm Dark Matter (WDM) scenarios gives constraints much stronger than the considered effect. Qualitatively, the dark atom model of dark matter is just a “warmer than cold dark matter” scenario.

4. Dark-Atom Interaction with Nuclei

4.1. Dark Atom Solution for the Puzzles of Direct Dark Matter Search

The proposal regarding the existence of O-helium atoms offers an explanation for the conflicting outcomes observed in experiments aimed at directly detecting dark matter. These discrepancies stem from the intricacies inherent in the interaction between dark atoms and the detecting material in underground detectors [70]. For instance, the affirmative findings from experiments like $DAMA/NaI$ and $DAMA/LIBRA$, suggesting the identification of dark matter particles, stand in contrast to the negative outcomes from other experiments such as $XENON100$, $LUX$, and $CDMS$, which also pursue direct detection of dark matter particles.

Initial qualitative assessments suggest that dark atoms possess the capability of forming low-energy bound states with nuclei of intermediate masses, while excluding such binding with heavy element nuclei [71]. Considering the scalar and isoscalar essence of XHe, the formation of such low-energy states of dark atoms with substance nuclei can solely occur through the electric dipole $E1$ transition, entailing isotopic invariance violation. This transition is directly proportional to the square of the relative velocity, hence temperature, and thus, it is attenuated in cryogenic experiments [7,71]. These observations elucidate the absence of positive outcomes in experiments other than $DAMA/NaI$ or $DAMA/LIBRA$, as their methodology primarily targets the detection of recoil nuclei effects, potentially attributing energy release effects to background events.

The retardation experienced by cosmic XHe particles as they traverse terrestrial matter presents a formidable obstacle to the direct detection of dark matter particles via methods reliant on recoiling effects stemming from WIMP–nucleus collisions. The parameters inherent in X-helium atoms facilitate the efficient deceleration of the flow of cosmic dark atoms to thermal energies and their gradual diffusion toward the Earth’s core. Consequently, the pursuit of dark matter particle detection through recoil nucleus effects becomes unfeasible in this scenario [62]. Hence, an alternative explanation is imperative to interpret the results of the $DAMA$ experiment, which we, in turn, advance and strive to substantiate.

When slow-moving X-helium atoms interact with nuclei, they possess the capability to establish low-energy bindings with one another. Within the margin of uncertainty nuclear physics parameters, there exists a distinct range wherein the binding energy within the OHe-$Na$ system aligns with the interval of 2–4 keV [7]. Upon the capture of dark atoms entering this bound state, they liberate energy. This released energy manifests as an ionization signal detectable by detectors such as $DAMA$.

The abundance of XHe within underground detectors hinges on the equilibrium between the influx of cosmic dark matter atoms and their diffusion towards the Earth’s center. The rapid regulation of X-helium presence within the Earth’s crust occurs due to the dynamics of the interplay between dark atoms and terrestrial matter, influenced by incoming cosmic XHe and variations in its flux. Consequently, the capture rate of dark atoms is expected to display annual fluctuations, which will be reflected in the annual modulation of the observed ionization signal generated by these interactions.

A significant implication of this proposed elucidation is the appearance of anomalous superheavy sodium isotopes within the detection material utilized in $DAMA/NaI$ or $DAMA/LIBRA$ experiments. Should these atypical isotopes remain partially ionized, their mobility is governed by atomic cross-sections and is approximately nine orders of magnitude lower compared to OHe [71]. Consequently, they will persist within the detector. Hence, employing mass spectroscopy to scrutinize the material of detectors can provide additional evidence regarding the presence of an O-helium component within the $DAMA$ signal. However, the techniques employed for this analysis must carefully account for the thin structure of the bound states of OHe-$Na$, since their binding energy merely spans a few keV [7].

Furthermore, when dark atoms interact with matter, especially within regions characterized by elevated concentrations of XHe in the Galactic center, it can lead to the excitation of X-helium. As a result, the surplus of the positron annihilation line detected by $INTEGRAL$ in the Galactic central region may be ascribed to the production of pairs emitted by the excited XHe dark atoms generated through these collisions [7].

4.2. The Problem of Potential Barrier in Dark-Atom Interaction with Nuclei

The XHe hypothesis is characterized by its simplicity since it is based on a single parameter of “new” physics—the mass of the X particle. Nevertheless, its successful application hinges upon a comprehensive grasp of known nuclear and atomic physics fields that have not yet been extensively applied to non-classical bound systems like XHe dark atoms.

Exploration into the active influence exerted by this form of dark matter on nuclear transformations is imperative, as it is pivotal for the validation of the dark atoms hypothesis and facilitates advancements in the area of nuclear physics pertaining to X-helium. Such investigations are particularly critical when evaluating the quantitative significance of dark atoms in primary cosmological nucleosynthesis and the evolutionary processes of stars [7,71].

The primary problem regarding XHe atoms lies in their potential for strong interactions with the nucleus of substance, stemming from the unshielded nuclear attraction between helium and matter nuclei. Such interaction bears the risk of destroying the bound structure of dark matter atoms, potentially leading to the formation of anomalous isotopes. Stringent experimental constraints are imposed on the concentration of these isotopes within terrestrial and aquatic environments [19]. To mitigate the excessive generation of anomalous isotopes, it is conjectured that the effective interaction potential between XHe and the nucleus of a heavy element should feature a shallow potential well and a dipole Coulomb barrier. This barrier is envisaged to impede the fusion of $He$ and/or the X particle with heavy nuclei, as illustrated in Figure 11. Such structure of the potential is conditioned by the competition between Coulomb repulsion (nucleus of heavy element–nHe) and nuclear attraction (nucleus of heavy element–nuclear shell of a dark atom).

The problem associated with describing the interaction of a dark atom with the nucleus of matter is a three-body problem, the exact analytical solution of which is missing. Therefore, in order to evaluate the physical meaning of the proposed scenario involving a dipole potential barrier and a shallow well within the effective interaction potential, it becomes imperative to devise a numerical approach. This approach must adequately take into consideration the intricate dynamics and the non-trivial nature inherent in this three-body problem, thereby making i the validation of the proposed scenario’s implementation possible.

In contrast to ordinary atoms, dark atoms comprise a leptonic core and a nuclear-interacting shell, making usual approximations of atomic physics ineffective. Therefore, we are consistently developing a correct and precise quantum mechanical description of this three-body system: a bound system of a dark atom and the external nucleus, adding the required effects and interactions to enhance the accuracy of the result, which allows us to clarify the most significant points and key aspects of the proposed dark atom scenario. Thus, this section of this article introduces the description of a numerical quantum mechanical approach to describe the interaction between XHe and the nucleus of a heavy element using a numerical model to reconstruct the shape of the corresponding effective interaction potential.

4.3. Towards Correct Quantum Mechanical Description of Dark Atom Nuclear Physics

4.3.1. Isolated Dark Atom of O-Helium

It is known that in an alternating electric field created by the external nucleus, a dark atom should experience the Stark effect, which leads to the polarization of OHe. This should cause dipole Coulomb repulsion of the dark atom from the nucleus and the formation of a bound state of O-helium with this nucleus, determined by the well in front of the dipole Coulomb barrier in the effective interaction potential.

In order to restore with great accuracy the form of the effective interaction potential in the OHe–nucleus system, it is necessary to accurately calculate the Stark potential, which determines the interaction of a polarized dark atom (OHe dipole) with a charged nucleus of a heavy element. The Stark potential should have a fairly strong influence on the depth of the potential well, which characterizes the low-energy bound state of OHe with the nucleus of matter, and on the height of the dipole Coulomb barrier, leading to the repulsion of the dark atom and the nucleus of matter and preventing their fusion in the corresponding effective interaction potential. To do this, it is necessary to quantum-mechanically calculate the dipole moments $\delta$ of a dark atom polarized under the influence of an alternating external electric field (Stark effect) since the shape of the Stark potential depends on the value of the dipole moment $\delta$ according to the following formula:

$$U_{St}=e{Z}_{He}{E}_{nuc}\delta ,$$

(62)

where $E_{nuc}$ is the strength of the external electric field created by the nucleus of the heavy element.

For an accurate quantum mechanical calculation of the dipole moment of a polarized dark matter atom, in addition to the wave functions of helium corresponding to its ground state in the OHe–nucleus system, it is also necessary to calculate the wave function of the ground state of helium in an isolated (non-polarized) OHe atom. Therefore, to begin with, we considered $\widehat{H_0}$, that is, the Hamiltonian of an isolated (not subject to external influences) OHe dark matter atom. Having represented the operator $\widehat{H_0}$ in the form of a matrix, using a difference scheme, the eigenvalues of the Hamilton operator were numerically calculated, which are equal to the energies of helium $E_{OHe}$ in an isolated atom OHe, and its eigenvectors, which are equal to the $\Psi$-functions of helium in O-helium, i.e., solve the following one-dimensional Schrodinger equation:

$$\widehat{H_0}\Psi \left(\overrightarrow{r}\right)=E_{OHe}\Psi \left(\overrightarrow{r}\right),$$

(63)

or presented in another form:

$${\Delta }_r\Psi \left(\overrightarrow{r}\right)+{\displaystyle \frac{2m_{He}}{{\hslash }^2}}\left(E_{OHe}+{\displaystyle \frac{4e^2}{r}}\right)\Psi \left(\overrightarrow{r}\right)=0,$$

(64)

where $\overrightarrow{r}$ is the radius vector of the helium nucleus in the coordinate system with the origin at the center of the $O^{--}$ particle.

Theoretical calculations show that the energy levels of helium in the dark atom of OHe, $E_{n_{OHe}}$ obey the following pattern, by analogy with the hydrogen atom:

$$E_{n_{OHe}}={\displaystyle \frac{-8m_{He}{\alpha }^2}{n^2}}\mathrm{MeV},$$

(65)

where n is any natural number, $\alpha$ is the fine structure constant, and $-8m_{He}{\alpha }^2\approx -1.6\mathrm{MeV}$ is the energy level of the ground state of helium in O-helium or $E_{1_{OHe}}$.

As a result of the numerical solution of the one-dimensional Schrodinger Equation (63) with the helium radius vector interval $r=|2.5\times {10}^{-12}\mathrm{cm}|$, the first three eigenvalues of the Hamilton operator $\widehat{H_0}$ were numerically obtained: $E_{1,2,3_{num}}=-1.585,-0.393,-0.042\mathrm{MeV}$. Theoretical calculation (65) of the first three helium energy levels in the dark atom of O-helium gives the following results: $E_{1,2,3_{OHe}}=-1.589,-0.397,-0.177\mathrm{MeV}$. As can be seen from the results obtained, the first two energy levels obtained using numerical calculations are consistent with the theoretical values up to hundredths of values after the decimal point. For a quantum mechanical numerical calculation of the dipole moment of polarized OHe, it is enough for us to know the wave function corresponding to the first level of helium energy in an isolated O-helium atom.

It is possible to construct a discrete spectrum of helium energy levels in the dark atom potential and plot the graphs of the squared modulus of the wave function corresponding to these energy levels (Figure 12, which shows the first three energy levels).

4.3.2. Quantum-Mechanical Description of the Three-Body Problem in the OHe-Nucleus System

The three-body problem we are considering is described by the system XHe–nucleus. First, let us consider a special case of this system, when the XHe dark atom is a hydrogen-like O-helium Bohr atom. The origin of the coordinate system we have chosen is at the center of the particle $O^{--}$, which interacts with the point-like nucleus of $He$ by the Coulomb force and forms a connected dark matter atom system with it. The dark atom of OHe is in an external inhomogeneous electric field created by a third particle, which is the nucleus with charge number $Z_{nuc}$, number of neutrons $N_{nuc}$ and mass number A. This nucleus is gradually approaching the dark atom system, participating with it in electrical and strong nuclear interactions.

The Hamiltonian for the helium nucleus can be represented as:

$$\widehat{H}=\widehat{H_0}+\widehat{U},$$

(66)

where $\widehat{H_0}$ is the Hamiltonian of an isolated (not subject to external influences) OHe atom of dark matter, and $\widehat{U}$ is the potential of the interaction of helium with the outer nucleus.

Let us introduce the following vectors: $\overrightarrow{r}$, ${\overrightarrow{R}}_{OA}$ and ${\overrightarrow{R}}_{HeA}$, where $\overrightarrow{r}$ is the vector of the mutual distance between the particle $O^{--}$ and the helium nucleus, ${\overrightarrow{R}}_{OA}$ is the radius vector of the outer nucleus, and ${\overrightarrow{R}}_{HeA}$ is the vector drawn from the center of $He$ to the center of the outer nucleus. These vectors are connected as follows:

$${\overrightarrow{R}}_{HeA}={\overrightarrow{R}}_{OA}-\overrightarrow{r}.$$

(67)

Let us write $\widehat{H_0}$ and $\widehat{U}$:

$$\widehat{H_0}=-{\displaystyle \frac{{\hslash }^2}{2m_{He}}}\Delta -{\displaystyle \frac{4e^2}{r}},$$

(68)

$$\widehat{U}=U_{Coulomb}\left(\right|{\overrightarrow{R}}_{OA}-\overrightarrow{r}\left|\right)+U_{Nuc}\left(\right|{\overrightarrow{R}}_{OA}-\overrightarrow{r}\left|\right),$$

(69)

where $U_{Nuc}\left(\right|{\overrightarrow{R}}_{OA}-\overrightarrow{r}\left|\right)$ is the potential energy of the nuclear interaction, which we first write as the Woods–Saxon potential, and $U_{Coulomb}\left(\right|{\overrightarrow{R}}_{OA}-\overrightarrow{r}\left|\right)$ is the potential of the Coulomb interaction between a point-like helium nucleus and a non-point-like nucleus of a heavy element.

The nuclear potential will be calculated depending on the distance between the neutron distribution surfaces of interacting nuclei, that is, $U_{Nuc}\left(\right|{\overrightarrow{R}}_{OA}-\overrightarrow{r}\left|\right)$ is determined by the following expression:

$$U_{Nuc}\left(\right|{\overrightarrow{R}}_{OA}-\overrightarrow{r}\left|\right)=-{\displaystyle \frac{U_0}{1+exp\left(\frac{|{\overrightarrow{R}}_{OA}-\overrightarrow{r}|-R_{N_{nuc}}-R_{N_{He}}}{p}\right)}},$$

(70)

where $R_{N_{nuc}}$ is the root-mean-square radius of the neutron distribution in the nucleus of the heavy element, $R_{N_{He}}$ is the root-mean-square radius of the neutron distribution in helium, $U_0$ is the depth of the potential well (for the sodium nucleus equal to approximately 43 MeV), p is the diffuseness parameter equal to approximately $0.55\mathrm{fm}$.

The radii $R_{N_{nuc}}$ and $R_{N_{He}}$ are calculated as follows [72]:

$$R_{N_{nuc,He}}=\sqrt{{\displaystyle \frac{3}{5}}{R}_{0N_{nuc,He}}^2+{\displaystyle \frac{7{\pi }^2}{5}}{a}_{N_{nuc,He}}^2}\sqrt{1+{\displaystyle \frac{5b_{nuc,He}^2}{4\pi }}}\mathrm{fm},$$

(71)

where $b_{nuc,He}$ is the deformation parameter of the matter nucleus and helium nucleus, respectively (for the sodium nucleus, the deformation parameter was made equal to $b_{Na}=0.447$, the helium nucleus was considered as spherically symmetric; therefore, the deformation parameter of the helium nucleus was made equal to zero), $R_{0N_{nuc,He}}$ is the half radius of the neutron distribution in the nucleus of matter and the helium nucleus, respectively, and it is determined through the number of neutrons N and the number of protons Z of the corresponding nucleus:

$$R_{0N_{nuc,He}}=0.953N_{nuc,He}^{1/3}+0.015Z_{nuc,He}+0.774\mathrm{fm},$$

(72)

and $a_{N_{nuc,He}}$ is a dimensional parameter that also depends on the number of protons Z and the number of neutrons N of the corresponding nucleus:

$$a_{N_{nuc,He}}=0.446+0.072{\displaystyle \frac{N_{nuc,He}}{Z_{nuc,He}}}\mathrm{fm}.$$

(73)

The potential of the Coulomb interaction between a point helium nucleus and a nucleus of matter with a radius equal to the root-mean-square radius of the proton distribution, $R_{p_{nuc}}$, $U_{Coulomb}\left(\right|{\overrightarrow{R}}_{OA}-\overrightarrow{r}\left|\right)$ is calculated using the following formula:

$$U_{Coulomb}\left(\right|{\overrightarrow{R}}_{OA}-\overrightarrow{r}\left|\right)=\left\{\begin{array}{cc}{\displaystyle \frac{2e^2{Z}_{nuc}}{|{\overrightarrow{R}}_{OA}-\overrightarrow{r}|}}\hfill & \mathrm{for}|{\overrightarrow{R}}_{OA}-\overrightarrow{r}|>R_{p_{nuc}},\hfill \\ {\displaystyle \frac{2e^2{Z}_{nuc}}{2R_{p_{nuc}}}}\left(3-{\displaystyle \frac{|{\overrightarrow{R}}_{OA}-\overrightarrow{r}{|}^2}{R_{p_{nuc}}^2}}\right)\hfill & \mathrm{for}|{\overrightarrow{R}}_{OA}-\overrightarrow{r}|<=R_{p_{nuc}},\hfill \end{array}\right.$$

(74)

where $R_{p_{nuc}}$ is calculated as follows [72]:

$$R_{p_{nuc}}=\sqrt{{\displaystyle \frac{3}{5}}{R}_{0p_{nuc}}^2+{\displaystyle \frac{7{\pi }^2}{5}}{a}_{p_{nuc}}^2}\sqrt{1+{\displaystyle \frac{5b_{nuc}^2}{4\pi }}}\mathrm{fm},$$

(75)

$R_{0p_{nuc}}$ is the half radius of the distribution of protons in the nucleus, calculated using the charge number and the number of neutrons in the nucleus of the heavy element:

$$R_{0p_{nuc}}=1.322Z_{nuc}^{1/3}+0.007N_{nuc}+0.022\mathrm{fm},$$

(76)

and $a_{p_{nuc}}$ is a dimensional parameter, also determined by the number of protons and the number of neutrons of the nucleus of the heavy element:

$$a_{p_{nuc}}=0.449+0.071{\displaystyle \frac{Z_{nuc}}{N_{nuc}}}\mathrm{fm}.$$

(77)

Consequently, the Hamiltonian $\widehat{H}$ for helium in the system OHe–nucleus depends on the radius vectors $\overrightarrow{r}$ and ${\overrightarrow{R}}_{OA}$. However, by fixing the value of ${\overrightarrow{R}}_{OA}$ and successively changing the position of the heavy nucleus, i.e., by varying the value of the vector ${\overrightarrow{R}}_{OA}$, we can obtain a set of Schrodinger equations depending only on $\overrightarrow{r}$, each of which corresponds to a certain position of the outer nucleus relative to the dark atom.

Thus, the following Schrodinger equation needs to be solved:

$$\widehat{H}\Psi \left(\overrightarrow{r}\right)=E\Psi \left(\overrightarrow{r}\right),$$

(78)

or expanding $\widehat{H}$ and performing some transformations:

$$\Delta \Psi \left(\overrightarrow{r}\right)+{\displaystyle \frac{2m_{He}}{{\hslash }^2}}\left(E+{\displaystyle \frac{4e^2}{r}}-U_{Coulomb}\left(\right|{\overrightarrow{R}}_{OA}-\overrightarrow{r}\left|\right)-U_N\left(\right|{\overrightarrow{R}}_{OA}-\overrightarrow{r}\left|\right)\right)\Psi \left(\overrightarrow{r}\right)=0.$$

(79)

Representing the Hamilton operator in the form of a matrix using a difference scheme, one can numerically calculate the eigenvalues of the operator $\widehat{H}$, which are equal to the energies of helium E in the system OHe–nucleus for each fixed position ${\overrightarrow{R}}_{OA}$ of the outer nucleus of matter. The eigenvectors of the Hamiltonian are also calculated, which are $\Psi$-functions of $He$ for a given system.

To do this, in addition to representing the Laplace operator in the form of a matrix, it is also necessary to reconstruct the matrix form of the potential in which the helium nucleus is located for each fixed value of ${\overrightarrow{R}}_{OA}$:

$$U_{He}=-{\displaystyle \frac{4e^2}{r}}+U_{Coulomb}\left(\right|{\overrightarrow{R}}_{OA}-\overrightarrow{r}\left|\right)+U_N\left(\right|{\overrightarrow{R}}_{OA}-\overrightarrow{r}\left|\right).$$

(80)

Figure 13 shows an example of the reconstructed total interaction potential of the helium in the system OHe–$Na$, $U_{He}$, depending on the radius of the helium vector $\overrightarrow{r}$ for a fixed value of the radius vector ${\overrightarrow{R}}_{OA}$ of the outer nucleus $Na$.

Figure 13 shows the Coulomb and nuclear interaction potentials between the helium and the sodium nucleus, as well as the Coulomb potential between the helium and the $O^{--}$ particle and the total interaction potential for the helium nucleus in the OHe–$Na$ system.

Thus, the quantum mechanical description and numerical solution of the three-body problem in the system OHe–nucleus is based on the numerical solution of the Schrodinger equations for the helium nucleus in the OHe–nucleus system for each fixed position ${\overrightarrow{R}}_{OA}$ of the outer nucleus, using the representation of the Hamilton operator of the helium nucleus in matrix form and the numerical calculation of its eigenvalues and eigenvectors, which are equal to the energies and $\Psi$-functions of helium in the OHe–nucleus system, respectively.

4.3.3. Solution of the Schrodinger Equations for $He$ in the OHe–Nucleus System

In the absence of the outer nucleus of matter, the dark matter atom is not polarized, and the ground state of helium in OHe has an energy level of approximately $1.6\mathrm{MeV}$. After the outer nucleus begins to approach the dark atom, it is polarized due to the influence of the alternating electric field created by the nucleus of the heavy element on it— the Stark effect occurs, OHe has a non-zero dipole moment, and the dark matter atom begins to interact with the nucleus of matter as a dipole. The corresponding interaction can be described using the Stark potential (see Formula (62)). The dark atom hypothesis assumes that in the effective potential of interaction between OHe and the nucleus of the heavy element, a dipole barrier should arise that will not allow dark matter atom particles to enter the nucleus, and a low-energy bound state will arise between the dark atom and the nucleus of the heavy element.

When solving the one-dimensional Schrodinger equation (SE) for a helium nucleus in the OHe–nucleus system (see Formula (79)), it is necessary to set the interval of the helium radius vector $\overrightarrow{r}$ at a fixed position of the nucleus ${\overrightarrow{R}}_{OA}$. Therefore, $\overrightarrow{r}$ is a free parameter, and the shape of the total potential in which the helium nucleus is located and in which the corresponding SE is solved at a fixed position of the nucleus of the heavy element depends on it. Since we need to solve the set of SE for each position of the nucleus of a substance gradually approaching the dark atom, it is necessary to set the interval in which the radius vector of the nucleus ${\overrightarrow{R}}_{OA}$ changes. If the interval $\overrightarrow{r}$ is set such that it completely overlaps with the interval ${\overrightarrow{R}}_{OA}$, the most probable position of helium when solving the SE will always be in the deep potential well created by the heavy nucleus. Since the initial position of the helium nucleus is known, it is located in a dark atom. Since the OHe atom is a bound quantum mechanical system even before the start of its interaction with the nucleus of matter, it is necessary to set such intervals of variation in $\overrightarrow{r}$ and ${\overrightarrow{R}}_{OA}$ so that their boundary points are close but do not overlap. This is necessary so that helium ends up in the dark atom and gradually begins to “feel” the approaching nucleus of the heavy element and, when its position is close, begins to tunnel into the nucleus through the Coulomb barrier with increasing probability. Therefore, for a given interval $\overrightarrow{r}$, the right and left ends of which are equal in magnitude and opposite in sign, the interval ${\overrightarrow{R}}_{OA}$ is specified such that its initial position is as far as possible from the dark atom, and the final position is close to the right end of the helium radius vector interval. The dark matter atom will experience greater polarization the closer it gets to the outer nucleus.

As the nucleus of the substance approaches, the ground state of the helium nucleus in O-helium will also change. To calculate the change in the dipole moment of a polarized dark atom, we need to know the change in the ground state of a polarized dark matter atom and the wave functions corresponding to these states. By gradually bringing the outer nucleus closer to OHe and solving the Schrodinger equation for helium for each corresponding fixed radius vector of the nucleus ${\overrightarrow{R}}_{OA}$, we calculated the entire energy spectrum of helium in a polarized dark atom. We have always considered the ${}^{11}\mathrm{Na}$ nucleus as the outer nucleus.

Figure 14 plots the dependence of the energy value of the ground state of helium in a polarized dark atom as a function of the radius vector of the outer nucleus of sodium at an interval of a helium radius vector equal to $r=|1.1\times {10}^{-12}\mathrm{cm}|$.

In Figure 14, the red stars show the energy values of the ground state of helium in the polarized OHe dark atom, corresponding to a fixed value of the radius vector of the sodium nucleus ${\overrightarrow{R}}_{OA}$. From Figure 14, it is clear that when the nucleus of matter is far from OHe, the dark matter atom can be considered isolated and the energy of the ground state of $He$ in the dark atom approaches the value $-1.6\mathrm{MeV}$, which corresponds to the binding energy of the O-helium atom. However, the closer the sodium nucleus comes to the O-helium, the more the OHe atom is polarized and the more the energy of the helium ground state is distorted, tending to values above zero. Further, when the sodium nucleus comes too close to the dark atom, helium begins to tunnel into the sodium nucleus with a high probability and the energy levels tend to the most probable value corresponding to the square of the modulus of the wave function with the highest probability of finding helium at the center of the sodium nucleus.

Therefore, having solved the set of Schrodinger equations for helium in the OHe–$Na$ system for various fixed positions of the nucleus of the heavy element ${\overrightarrow{R}}_{OA}$ relative to the dark atom, we obtained a spectrum of energy values for the ground state of helium in a polarized OHe atom and the wave functions corresponding to these states of helium.

Further, using the normalized wave function of the ground state of helium in an unpolarized dark atom, ${\Psi }_{OHe}$, restored by solving the Schrodinger equation for helium in an isolated OHe atom and the normalized wave functions of helium in a polarized dark atom, corresponding to different values of the ground energy state, ${\Psi }_{OHeNa}$, we calculated the spectrum of values of the dipole moment $\delta$ of polarized OHe. The value of the dipole moment $\delta$ corresponding to a certain ${\Psi }_{OHeNa}$ is calculated using the following formula:

$$\delta ={\int }_r{\Psi }_{OHe}^{*}·r·{\Psi }_{OHeNa}·4\pi r^{2}dr.$$

(81)

In order to calculate the integral (81), it is necessary to find its limits of integration. Since we are calculating the dipole moments of a polarized dark atom, we need to use the probabilities of finding helium in the region inside the dark atom. In order to determine the left and right boundaries of integration, it is necessary to find the intersection points of the graph of the squared modulus of the helium wave function and the graph of the total potential of helium in the OHe–$Na$ system at a fixed value of ${\overrightarrow{R}}_{OA}$. That is, for each fixed position of the nucleus of a heavy element, it is necessary to determine the region of integration inside the dark atom or, in other words, the boundaries of integration, which are the points of intersection of the graphs of the total potential of helium and its squared modulus of the wave function corresponding to a certain value of the ground state of energy.

Figure 15 shows one example of determining the boundary of integration for calculating the integral (81). The blue solid line shows the total potential of helium in the OHe–$Na$ system for a fixed position of sodium ${\overrightarrow{R}}_{OA}$, the red solid line shows the graph of the squared modulus of the wave function of the ground state of helium in polarized dark atom with fixed ${\overrightarrow{R}}_{OA}$, black circles show the points of intersection of two graphs. The first two points of intersection of the graphs from left to right are taken as the integration boundary for (81). The case depicted in Figure 15 shows the state when the dark atom is negatively polarized since the probability of finding helium to the left of the origin or of the $O^{--}$ particle is greater than to the right. The sodium nucleus has already come close enough for the nuclear potential to begin to be “perceptible” (a potential well of the heavy nucleus begins to appear to the right of the Coulomb barrier), but not enough for effective penetration of helium through the potential barrier to begin.

After calculating the spectrum of values of the dipole moment $\delta$ of polarized OHe, corresponding to different positions of the nucleus of the sodium ${\overrightarrow{R}}_{OA}$, it is possible to plot the dependence of the dipole moment of a polarized dark atom on the radius vector of the sodium nucleus ${\overrightarrow{R}}_{OA}$ (see Figure 16).

In Figure 16, the red stars show the values of the dipole moments of the polarized OHe atom, corresponding to fixed values of the radius vector of the sodium nucleus ${\overrightarrow{R}}_{OA}$ for the helium radius vector interval $r=|1.1\times {10}^{-12}\mathrm{cm}|$. From Figure 16, it is clear that when the nucleus of a heavy element is far from OHe, the dark matter atom can be considered isolated and the value of the dipole moment tends to zero. The closer the sodium nucleus comes to the O-helium, the more strongly the OHe atom is polarized and the more the dipole moment increases in magnitude, becoming more negative. Sodium increasingly repels helium due to the Coulomb force, and therefore, it is more likely to take a position more to the left of the $O^{--}$ particle. After the sodium nucleus approaches sufficiently close to the dark atom and the value of ${\overrightarrow{R}}_{OA}$ becomes close to the right boundary point of the helium radius vector interval $\overrightarrow{r}$, the nuclear force between helium and sodium begins to dominate over their electrical Coulomb interaction, so $\delta$ begins to tend to zero.

At some point, a sufficiently high probability of helium tunneling into the nucleus of the heavy element appears, and the dipole moment changes sign and becomes positive (see Figure 17). At the same time, up to a certain point, the probability of finding helium inside the dark atom remains non-zero, which means that the O-helium atom has not yet collapsed, and the dark atom has changed polarization, that is, the helium has taken position between the particle $O^{--}$ and the nucleus of matter. Therefore, the dipole Coulomb repulsion of helium and the sodium nucleus becomes possible due to the dipole interaction between OHe and the nucleus of the heavy element, due to which a low-energy bound state of the OHe-$Na$ system can be formed.

The change in the sign of the dipole moment is consistent with the theory since, when the heavy nucleus is far from the dark atom, helium is repelled from it more strongly than it is attracted because the Coulomb interaction is stronger than the nuclear interaction at such distances, and the $O^{--}$ particle, on the contrary, is attracted and becomes closer to the sodium nucleus due to the Coulomb force. However, when the sodium nucleus comes close enough to OHe, the nuclear interaction begins to dominate, and it is assumed that helium should take a position between the $O^{--}$ particle and the sodium nucleus, which manifests itself in a change in the sign of the dipole moment. However, due to the dipole Coulomb barrier, helium must again be repelled from the sodium nucleus, which should prevent the destruction of the dark atom. As we further decrease ${\overrightarrow{R}}_{OA}$, that is, as sodium approaches the dark atom, the positive value of the dipole moment begins to decrease and tends to zero (see Figure 17). This occurs due to the increasingly deeper penetration of helium into the sodium nucleus and an increase in the probability of tunneling, while the probability of finding helium inside the dark atom tends to zero and the values of the dipole moment also become zero.

Figure 18 shows the first few graphs of the squared modulus of wave functions (red solid line) of the energy values of the ground state of helium in the total potential of the OHe–$Na$ system (blue solid line) corresponding to certain positions of the sodium nucleus relative to the dark atom at the moment when repolarization of the dark atom begins to occur, that is, when the dipole moment from the maximum negative value begins to tend toward a value above zero and some probability of helium tunneling into the sodium nucleus through the Coulomb barrier appears. At the end of repolarization, the dipole moment changes sign and becomes positive. This process is shown in the corresponding graphs of the dependence of the dipole moment on the radius vector of the sodium nucleus presented in Figure 16 and Figure 17.

Figure 19 shows several graphs of the squared modulus of wave functions (red solid line) of the energy values of the ground state of helium in the total potential of the OHe–$Na$ system (blue solid line) corresponding to certain positions of the sodium nucleus relative to the dark atom at the moments when helium begins to tunnel with a very high probability from the repolarized dark atom into the sodium nucleus, that is, when the dipole moment from the maximum positive value begins to tend to zero and the probability of finding helium in a dark atom also becomes practically zero. This can be seen in the graph of the dependence of the dipole moment on the radius vector of the sodium nucleus shown in Figure 17.

Using the values of the dipole moment calculated quantum mechanically, we can restore the shape of the Stark potential (see Formula (62)), which characterizes the electrical interaction of a polarized dark atom with the nucleus of matter. In order to restore the total effective interaction potential in the OHe–$Na$ system, it is also necessary to restore the form of the nuclear interaction potential of the Woods–Saxon type between helium and sodium nuclei, and the form of the electrical interaction potential of an unpolarized OHe dark atom with a sodium nucleus $U_{XHe}^e$, obtained by solving the self-consistent Poisson equation, taking into account the effect of screening the $He$ nucleus by the $O^{--}$ particle, which manifests itself only at close distances from dark atom, since it decays exponentially (see Section 5.1—Approach of Reconstructing of Interaction Potentials in the XHe–nucleus system in [63]). As a result, summing up all the listed potentials, the form of the effective potential of the interaction of the OHe dark atom with the sodium nucleus has been restored (see Figure 20).

The total effective potential of the interaction of O-helium with the nucleus of the sodium, depicted by the red dotted line in Figure 20, can be interpreted as the potential in which the sodium nucleus is located in the OHe–$Na$ system. On the chosen scale, the nuclear potential of the Woods–Saxon type (green dotted line) and $U_{XHe}^e$ (blue dotted line) practically does not differ from zero, and the potential well in the total effective interaction potential is completely determined by the Stark potential (gray dotted line) and thus the negative values of the dipole moment of polarized OHe. However, this potential well is quite deep, about $0.5\mathrm{MeV}$, so the bound state of $Na$ with a dark atom in this potential corresponding to certain manually specified intervals $\overrightarrow{r}$ and ${\overrightarrow{R}}_{OA}$ will also be high-energy, which is very different from the result of the $DAMA$ experiment. It is also possible to depict in more detail the effective total potential of interaction of O-helium with a sodium nucleus near the region corresponding to the moment of repolarization of the dark atom, that is, the moment when the dipole moment takes positive values (see Figure 21).

From Figure 21, it is clear that the theoretically assumed presence of a positive dipole Coulomb barrier in the total effective interaction potential for the sodium nucleus in the OHe–$Na$ system depends on whether the positive potential barrier in the Stark potential (gray dotted line), determined by the positive values of the dipole moment of the repolarized dark atom, “overcome” negative values of the nuclear interaction potential (green dotted line) and the electrical interaction potential of the unpolarized OHe dark atom with a sodium nucleus $U_{XHe}^e$ (blue dotted line).

According to Figure 21, when the distance between helium and the sodium nucleus is about $11.6\mathrm{fm}$, the maximum positive value of the Stark potential turns out to be less than the negative value of the sum of the nuclear potential and the potential $U_{XHe}^e$, which leads to negative values of the total effective interaction potential. However, since the interval of the radius vector of the helium nucleus $\overrightarrow{r}$, in which the Schrodinger equation for helium in the OHe–$Na$ system is solved at a fixed position of the nucleus of the heavy element ${\overrightarrow{R}}_{OA}$, is a free parameter and is set manually, we can increase the $\overrightarrow{r}$ interval and, by appropriately changing the ${\overrightarrow{R}}_{OA}$ interval, make the helium in the dark atom begin to “feel” the influence of the sodium nucleus at greater distances than has been done so far, since the final position of the sodium nucleus ${\overrightarrow{R}}_{OA}$ is given close to the right boundary of the helium radius vector interval $\overrightarrow{r}$. This will lead to the fact that the maximum negative value of the dipole moment of the polarized dark atom will decrease since it is largely determined by the Coulomb interaction between the helium and sodium nuclei, which decreases with increasing distance between the nuclei on scales larger than the radius of the sodium nucleus. Consequently, the maximum negative value of the Stark potential, which determines the depth of the potential well in the total effective interaction potential of the OHe–$Na$ system, will also decrease. In addition, an increase in the $\overrightarrow{r}$ interval and the corresponding change in the ${\overrightarrow{R}}_{OA}$ interval, in turn, will also affect the width of the Coulomb barrier in the total potential for helium in the OHe–$Na$ system. It will become much wider, which will ultimately affect the wave functions of helium and reduce the probability of its tunneling into the nucleus of a heavy element. Consequently, this should also lead to a decrease in the positive values of the dipole moment of the repolarized dark atom and, therefore, a decrease in the positive potential barrier in the Stark potential. However, at the same time, due to an increase in the distance between interacting nuclei, the values of the nuclear potential and the $U_{XHe}^e$ potential will also decrease, which can lead to the expected positive values of the potential barrier in the total effective interaction potential for sodium in OHe–$Na$ system.

In the OHe hypothesis, it is expected that the magnitude of the dipole Coulomb barrier in the total effective interaction potential of the OHe–$Na$ system will make it possible to avoid the direct fusion of the sodium nucleus with the dark atom. The relative velocity of the sodium nucleus in the OHe–$Na$ system, under the conditions of the $DAMA$ experiment, is thermal and approximately corresponds to the values of normal room temperature (∼$300\mathrm{K}$), that is, the sodium nucleus in the OHe–$Na$ system will move with kinetic energy approximately equal to ∼$2.6\times {10}^{-2}\mathrm{eV}$. Therefore, it is expected that the height of the dipole Coulomb barrier in the total effective interaction potential of the OHe–$Na$ system will be greater than the kinetic energy of sodium.

Thus, increasing the interval of the helium radius vector from $r=|1.1\times {10}^{-12}\mathrm{cm}|$ up to $r=|2.5\times {10}^{-12}\mathrm{cm}|$ and taking the interval $R_{OA}=[3.5;2.8]\times {10}^{-12}\mathrm{cm}$, we solved the one-dimensional Schrodinger equations for helium in the OHe–$Na$ system for each fixed value of ${\overrightarrow{R}}_{OA}$, we also calculated the dipole moment values for each value of the energy of the ground state of helium in a polarized dark atom and restored the shape of the total effective potential of interaction between OHe and the sodium nucleus (see Figure 22 and Figure 23).

From Figure 22, it follows that the depth of the potential well in the total effective potential of the interaction of the dark atom with the nucleus of the sodium (red dotted line), as expected, has decreased. Comparing Figure 22 with Figure 20 one can see that the depth of the potential well after increasing the interval $\overrightarrow{r}$ and increasing the values of the position of the sodium nucleus relative to the dark atom ${\overrightarrow{R}}_{OA}$, decreased from ∼$0.5\mathrm{MeV}$ to ∼$35\mathrm{keV}$.

In Figure 23, one can see that at distances between the helium nucleus and the sodium nucleus corresponding to approximately ∼$28.3\mathrm{fm}$, the nuclear potential of the Woods–Saxon type (green dotted line) and the potential of the electrical interaction of unpolarized OHe dark atom with the sodium nucleus $U_{XHe}^e$ (blue dotted line) are practically equal to zero, and as expected, a positive potential barrier appears in the total effective interaction potential of the OHe–$Na$ system, which is completely determined by the Stark potential. At the same time, from a comparison of Figure 21 with Figure 23, it is clear that, as expected, after increasing the interval $\overrightarrow{r}$ and increasing the values of the position of the sodium nucleus relative to the dark atom ${\overrightarrow{R}}_{OA}$, the positive potential barrier in the Stark potential decreased from ∼$7.5\mathrm{eV}$ to ∼$0.25\times {10}^{-2}\mathrm{eV}$ due to a decrease in the magnitude of the positive values of the dipole moment of the repolarized dark atom.

Thus, the dipole Coulomb barrier in the total effective interaction potential of the OHe–$Na$ system in Figure 23 becomes less than the kinetic energy with which the sodium nucleus in the OHe–$Na$ system should move towards the dark atom under the conditions of the $DAMA$ experiment (∼$2.6\times {10}^{-2}\mathrm{eV}$) approximately 10 times. However, the depth of the potential well in the total effective potential of the interaction of a dark atom with the nucleus of the sodium is approximately six times greater than the values obtained in the $DAMA$ experiment ($6\mathrm{keV}$) and even more than the expected theoretical energy values of the low-energy bound state of sodium with a dark atom (∼$4\mathrm{keV}$). Therefore, a further increase in the interval of the helium radius vector $\overrightarrow{r}$, within which the Schrodinger equation for helium in the OHe–$Na$ system is solved and the values of the position of the sodium nucleus are relative to the dark atom ${\overrightarrow{R}}_{OA}$, will reduce the depth of the potential well in the total effective interaction potential for sodium in the OHe–$Na$ system, and will also further reduce the height of the dipole Coulomb barrier, which is already less than the expected values. Consequently, it is necessary to improve the accuracy of our numerical model, for which we need to take into account additional, previously unaccounted-for effects that characterize the interaction of the dark atom with the nucleus of matter. Such an effect, for example, is the centrifugal potential of interaction between the dark atom and the nucleus of the sodium $U_{ro{t}_{(OHe-Na)}}$.

4.3.4. Adding Centrifugal Potential to the Quantum Mechanical Numerical Model of the OHe–Nucleus System

The centrifugal potential of the interaction of the OHe dark atom with the nucleus of the sodium $U_{ro{t}_{(OHe-Na)}}$, which depends on the total angular momentum of the system of interacting particles, ${\overrightarrow{J}}_{(OHe-Na)}$, and on the distance between interacting particles, R, without taking into account the moments of inertia of nuclei, is determined as follows (see Formula (27) of article [73]):

$$U_{ro{t}_{(OHe-Na)}}\left(R\right)={\displaystyle \frac{{\hslash }^2{c}^2{J}_{(OHe-Na)}(J_{(OHe-Na)}+1)}{2\mu c^2{R}^2}},$$

(82)

where $\mu$ is the reduced particle mass.

Since the mass of OHe is completely determined by the mass of the heavy particle $O^{--}$, which we took as equal to $1\mathrm{TeV}$, and the mass of the sodium nucleus, approximately equal to $m_{Na}\approx 21.4\mathrm{GeV}$, which is much less than the mass of OHe, the reduced mass is approximately equal to the mass of sodium $\mu \approx m_{Na}/c^2$.

The total angular momentum of interacting particles ${\overrightarrow{J}}_{(OHe-Na)}$ can be calculated as follows:

$${\overrightarrow{J}}_{(OHe-Na)}\left(\rho \right)={\overrightarrow{l}}_{(OHe-Na)}\left(\rho \right)+{\overrightarrow{I}}_{Na}+{\overrightarrow{I}}_{OHe},$$

(83)

where ${\overrightarrow{l}}_{(OHe-Na)}\left(\rho \right)$ is the orbital angular momentum of the interacting particles depending on the impact parameter $\rho$, ${\overrightarrow{I}}_{Na}$ is the intrinsic angular momentum of the sodium nucleus, and ${\overrightarrow{I}}_{OHe}$ is the spin of the O-helium dark atom. ${\overrightarrow{I}}_{OHe}$ is equal to the sum of the $O^{--}$ particle spin, ${\overrightarrow{I}}_{O^{--}}$, and the intrinsic angular momentum of the helium nucleus, ${\overrightarrow{I}}_{He}$:

$${\overrightarrow{I}}_{OHe}={\overrightarrow{I}}_{He}+{\overrightarrow{I}}_{O^{--}}.$$

(84)

We consider the case of a head-on collision between a sodium nucleus and OHe. When the impact parameter of the sodium nucleus swooping on the dark atom is zero, $\rho =0$, the orbital momentum of the interacting particles is also zero ${\overrightarrow{l}}_{(OHe-Na)}\left(0\right)=\overrightarrow{0}$. The intrinsic angular momentum of the sodium nucleus is equal to ${\overrightarrow{I}}_{Na}=\overrightarrow{3/2}$.

The spin of the OHe dark atom is determined by the spin of the $O^{--}$ particle since the intrinsic angular momentum of the helium nucleus is equal to ${\overrightarrow{I}}_{He}=0$. The spin of the $O^{--}$ particle, ${\overrightarrow{I}}_{O^{--}}$, is a model-dependent parameter since it depends on the structure of the $O^{--}$ particle. In models with four or five generations of fermions, the existence of a stable state with a charge of $-2$, ${\Delta }_{\overline{U}\overline{U}\overline{U}}^{--}$, consisting of three anti-quarks, for example, fourth generation $\overline{U}$: ${\Delta }_{\overline{U}\overline{U}\overline{U}}^{--}=\left(\overline{U}\overline{U}\overline{U}\right)$ [74]. Thus, if $O^{--}$ is ${\Delta }_{\overline{U}\overline{U}\overline{U}}^{--}$ and consists of quarks of new families, then ${\overrightarrow{I}}_{O^{--}}=\overrightarrow{3/2}$. The walking technicolor (WTC) model suggests the existence of a new type of interaction within the framework of SU(2) symmetry, which binds a new type of quark [8,13]. Technibaryons are considered within the framework of the technicolor model. Technibaryons are particles formed from techniquarks that have their own type of interaction (not manifested explicitly at energies below the technicolor confinement scale) and charge. If the $O^{--}$ particle is technibaryon, then ${\overrightarrow{I}}_{O^{--}}=\overrightarrow{0}$ or $\overrightarrow{1}$. The WTC also introduces the existence of the fourth generation of technileptons. If the $O^{--}$ particle is technilepton, then ${\overrightarrow{I}}_{O^{--}}=\overrightarrow{1/2}$.

Thus, in the case we are considering, when the impact parameter $\rho =0$, the total angular momentum of the interacting O-helium dark atom and sodium nucleus, ${\overrightarrow{J}}_{(OHe-Na)}$, is calculated as follows:

$${\overrightarrow{J}}_{(OHe-Na)}={\displaystyle \frac{\overrightarrow{3}}{2}}+{\overrightarrow{I}}_{O^{--}}.$$

(85)

Before adding the centrifugal potential $U_{ro{t}_{(OHe-Na)}}$ to the total effective potential of interaction between OHe and the nucleus of the sodium in the OHe–$Na$ system, it is also necessary to take into account the centrifugal potential of the interaction between the helium nucleus and the sodium nucleus $U_{ro{t}_{(He-Na)}}$ when solving the one-dimensional Schrodinger equation for $He$ in the OHe–$Na$ system. That is, it is necessary to add $U_{ro{t}_{(He-Na)}}$ to the total potential in which the helium nucleus is located in the OHe–$Na$ system (see Formula (80)) for a fixed position of the nucleus of a heavy element ${\overrightarrow{R}}_{OA}$:

$$U_{He+rot}=-{\displaystyle \frac{4e^2}{r}}+U_{Coulomb}\left(\right|{\overrightarrow{R}}_{OA}-\overrightarrow{r}\left|\right)+U_N\left(\right|{\overrightarrow{R}}_{OA}-\overrightarrow{r}\left|\right)+U_{ro{t}_{(He-Na)}}\left(\right|{\overrightarrow{R}}_{OA}-\overrightarrow{r}\left|\right),$$

(86)

where $U_{ro{t}_{(He-Na)}}\left(\right|{\overrightarrow{R}}_{OA}-\overrightarrow{r}\left|\right)$ is calculated using the following formula:

$$U_{ro{t}_{(He-Na)}}\left(\right|{\overrightarrow{R}}_{OA}-\overrightarrow{r}\left|\right)={\displaystyle \frac{{\hslash }^2{c}^2{J}_{(He-Na)}(J_{(He-Na)}+1)}{2m_{He}{c}^2{|{\overrightarrow{R}}_{OA}-\overrightarrow{r}|}^2}},$$

(87)

where ${\overrightarrow{J}}_{(He-Na)}$ is the total angular momentum of the interacting helium and sodium nuclei.

${\overrightarrow{J}}_{(He-Na)}$ is equal to the intrinsic angular momentum of the sodium nucleus ${\overrightarrow{I}}_{Na}=\overrightarrow{3/2}$, since the intrinsic angular momentum of the helium nucleus ${\overrightarrow{I}}_{He}=\overrightarrow{0}$, and we consider the case of a zero impact parameter of a sodium nucleus incident on helium; therefore, the orbital momentum of the interacting helium and sodium nuclei is also zero. Thus, ${\overrightarrow{J}}_{(He-Na)}=\overrightarrow{3/2}$.

Figure 24 shows an example of the total interaction potential of helium in the OHe–$Na$ system, taking into account the centrifugal potential of interaction between the helium nucleus and the sodium nucleus, $U_{He+rot}$, depending on the radius vector of the helium $\overrightarrow{r}$ at a fixed value of the radius vector ${\overrightarrow{R}}_{OA}$ of the outer nucleus of $Na$.

Figure 24 shows the potentials of the Coulomb interaction and the nuclear interaction between helium and the nucleus of the heavy element and also shows the centrifugal potential of the interaction between the helium nucleus and the sodium nucleus at zero impact parameter $U_{ro{t}_{(He-Na)}}\left(\right|{\overrightarrow{R}}_{OA}-\overrightarrow{r}\left|\right)$; in addition, the Coulomb potential between helium and the $O^{--}$ particle and the total interaction potential for the helium nucleus in the OHe–$Na$ system are shown.

As a result, having solved the one-dimensional Schrodinger equations for helium in the $U_{He+rot}$ potential of the OHe–$Na$ system for various fixed positions of the sodium nucleus ${\overrightarrow{R}}_{OA}$, taking the interval $R_{OA}=[3.3;2.3]\times {10}^{-12}\mathrm{cm}$ with the helium radius vector interval equal to $r=|2.5\times {10}^{-12}\mathrm{cm}|$, we have obtained the energy spectrum of the ground state of helium in the polarized OHe atom and the wave functions corresponding to these helium states. Then we calculated the values of dipole moments for each value of the energy of the ground state of helium in a polarized dark atom. As a result, the form of the total effective interaction potential of OHe with the sodium nucleus in the OHe–$Na$ system was restored, taking into account the centrifugal interaction potential $U_{ro{t}_{(OHe-Na)}}$ for two values of the total angular momentum of interacting O-helium and sodium nucleus particles, ${\overrightarrow{J}}_{(OHe-Na)}$, at ${\overrightarrow{I}}_{O^{--}}=\overrightarrow{0}$ and $\overrightarrow{3/2}$, that is, for ${\overrightarrow{J}}_{(OHe-Na)}=\overrightarrow{3/2}$ and $\overrightarrow{3}$, respectively (see Figure 25 and Figure 26).

From Figure 25 and Figure 26, it is clear that the greater the value of the spin ${\overrightarrow{I}}_{O^{--}}$ and the total angular momentum of the interacting dark atom and sodium nucleus particles ${\overrightarrow{J}}_{(OHe-Na)}$, the more the centrifugal interaction potential $U_{ro{t}_{(OHe-Na)}}$ affects the total effective potential of interaction of OHe with the sodium nucleus, reducing the depth of the potential well, increasing the positive potential barrier, and preventing the free fusion of the substance nucleus with the dark atom and its destruction.

Since we are considering the three-body problem, we should take into account feedback between interacting particles, that is, the reverse impact of the results of interaction on its further course. Therefore, at the beginning, we consider the helium nucleus in the OHe–$Na$ system and describe the forces acting on helium, that is, we restore the total potential for helium in the OHe–$Na$ system and solve the Schrödinger equation for it in order to calculate the values of the dipole moment of the polarized dark atom for each fixed position of the sodium nucleus ${\overrightarrow{R}}_{OA}$. Thus, the influence of the sodium nucleus on the polarization of the dark atom, characterized by the magnitude of the dipole moment, is taken into account. Therefore, we, among other things, took into account the centrifugal interaction potential between the helium nucleus and the sodium nucleus $U_{ro{t}_{(He-Na)}}$, which affects the shape of the total interaction potential of helium in the OHe–$Na$ system, $U_{He+rot}$, and through the influence on $U_{He+rot}$ influences the values of the dipole moment. This can be seen, for example, from Figure 26, which, when comparing with Figure 22, one can notice that the depth of the potential well in the Stark potential, which in Figure 22 practically coincides with the total effective potential of interaction between OHe and the sodium nucleus, increased in Figure 26 after adding $U_{ro{t}_{(He-Na)}}$ to the Schrödinger equation. This means that the $U_{ro{t}_{(He-Na)}}$ potential influenced the negative values of the dipole moment of the polarized dark atom. Also, the addition of $U_{ro{t}_{(He-Na)}}$ to the total helium interaction potential $U_{He+rot}$ affected the height of the dipole Coulomb barrier in the Stark potential, slightly reducing it, since $U_{ro{t}_{(He-Na)}}$ increases the positive potential barrier in $U_{He+rot}$ and reduces the probability of helium tunneling into the nucleus of the heavy element. Thus, the influence of the sodium nucleus on the polarization of OHe is first taken into account, after which the already polarized dark atom influences the sodium nucleus, interacting with it as a dipole. This interaction is characterized by the Stark potential. In addition, sodium interacts with the dark atom as a whole also through the centrifugal interaction potential $U_{ro{t}_{(OHe-Na)}}$, through the nuclear potential and $U_{XHe}^e$, which practically do not appear on large-distance scales between interacting particles.

As a result, we obtain the total effective potential of the interaction between OHe and the nucleus of the sodium in the OHe–$Na$ system, shown, for example, in Figure 26. Its shape depends on the spin value of the $O^{--}$ particle, $I_{O^{--}}$, but in any case, the shape of the total effective potential of the interaction of sodium with OHe qualitatively coincides with the theoretically expected one. This makes it possible to increase the interval of the helium radius vector $\overrightarrow{r}$ to obtain, corresponding to the experimental and theoretical expectations, a value of the potential well depth of about ∼$6\mathrm{keV}$ and a positive potential barrier with a height greater than zero and greater than the thermal kinetic energy of sodium ∼$2.6\times {10}^{-2}\mathrm{eV}$. This value of the positive potential barrier will prevent the fusion of $He$ and/or $O^{--}$ with the nucleus of matter and the destruction of the dark atom, which is a very important condition for the continued existence of the OHe dark atom hypothesis.

4.3.5. Discussion

This section of the article examines the hypothesis of composite dark matter, in which hypothetical stable particles with a charge of $-2n$ form neutral atom-like states of XHe with helium nuclei formed as a result of primary nucleosynthesis. Such hypothetical X-helium dark atoms of composite dark matter should interact with the nuclei of ordinary matter. The specifics of the interaction of XHe with the substance of underground detectors can explain the contradictory results of experiments in the direct search for dark matter particles. It is assumed that the interaction of slow X-helium dark atoms with the nuclei of substance can lead to the formation of a low-energy XHe–nucleus bound state. Within the uncertainty of nuclear physics parameters, there is a range in which the binding energy in the OHe-$Na$ system is in the range of 2–4 keV [7], which is a rather subtle effect. The consequence of the capture of dark atoms into this bound state is a corresponding release of energy, observed as an ionization signal in the detector.

To avoid the problem of overproduction of anomalous isotopes, it is assumed that the effective interaction potential between XHe and the nucleus of a heavy element will have a barrier that prevents the fusion of $He$ and/or X with the nucleus and the destruction of the dark atom, which is vital for the continued existence of the XHe dark atom hypothesis. To study the features and nuances of the interaction of the X-helium atom of composite dark matter with the nucleus of a substance, we are faced with the task of restoring the exact form of their effective interaction potential. This problem is a three-body problem and therefore does not have an exact analytical solution. Consequently, our work proposes a numerical modeling approach to accurately study the specifics of the interaction of dark atoms of dark matter with the nucleus of a heavy element in order to restore the exact form of the corresponding effective interaction potential and explain the results of experiments on the direct search for dark matter particles.

Such a numerical model is built in quantum mechanical approximation. This model describes system of three particles interacting with each other through electrical, centrifugal and nuclear forces. The quantum mechanical approach involves solving the Schrodinger equations for helium in the OHe–$Na$ system for each position of the nucleus relative to the O-helium atom, taking into account the features of nuclear and electromagnetic interactions characteristic of this system in order to calculate the polarization of dark matter atoms using a quantum mechanical method, calculate the dipole moments of polarized OHe atoms depending on the distance between the dark atom and the nucleus of the sodium, and thus, accurately restore the Stark potential, which significantly affects the shape of the effective interaction potential in the OHe–$Na$ system.

We have reconstructed the form of the helium interaction potential in the OHe–$Na$ system and solved the Schrodinger equations for the helium nucleus in isolated unpolarized dark atoms and in polarized O-helium in the OHe–$Na$ system. The dipole moments of the polarized OHe atom were calculated using the reconstructed wave functions of helium in isolated dark matter atoms and in the OHe–$Na$ system. The Stark potential has been calculated and the total effective interaction potential of the sodium nucleus with the OHe dark atom in the OHe–$Na$ system has been constructed, in which, in addition to the Stark potential, the nuclear potential, centrifugal potential and potential of electrical interaction of unpolarized OHe dark atoms with the sodium nucleus $U_{XHe}^e$.

In the future, to improve the accuracy of the results of restoring the effective interaction potential for the most physically correct description of the interaction of a dark atom with the nucleus of a heavy element and explain the results of experiments on the direct search for dark matter particles, it is planned to refine the quantum mechanical approach to restoring the effective interaction potential. It is planned to restore the nuclear and electromagnetic potentials of the interaction of X-helium with the nucleus of substance, taking into account the non-point nature of the interacting particles, namely, taking into account the distributions of electric charge and nucleons in the nuclei, and it is also planned to take into account the deformation of nuclei by considering a spherically asymmetric nucleus. Ultimately, after restoring the total effective interaction potential in the XHe-nucleus system, it is planned to solve the Schrodinger equation for the nucleus of a heavy element in this potential in order to study the possibilities of explaining the paradoxes of direct searches for dark matter particles and justifying physical experiments to test the hypothesis of dark atoms.

5. Conclusions

During the last three decades, the mainstream studies of physics beyond the Standard Model have been concentrated on the search for supersymmetric particles at the LHC and for WIMPs in underground direct dark matter searches. These trends were motivated by the necessity to solve the problems of the Standard Model by the involvement of SUSY particles, while the lightest SUSY particle was expected to possess WIMP properties nicely fitting the WIMP miracle in the explanation of cosmological dark matter. The lack of positive evidence for SUSY particles at the LHC and controversial published results of direct dark matter researche stimulate non-supersymmetric solutions for the problems of the Standard Model and the physical nature of dark matter.

The problems of divergence of the Higgs boson mass and of the origin of the electroweak energy scale may find solutions in the composite nature of the Higgs boson, while bound states of the Higgs boson constituents may lead to stable multiple-charged particles. Nontrivial electroweak SU(2) properties of such particles may provide a balance of excessive $-2n$ charged particles and baryon asymmetry, while binding of these excessive negatively charged particles with primordial helium nuclei can result in dark atom formation. It leads to the dark atom model of cosmological dark matter with the warmer-than-cold scenario of the large-scale structure formation. The nuclear interacting character of dark atoms makes their direct search by nuclear recoil in underground experiments impossible, but the existence of a few keV bound states of dark atoms with sodium nuclei and annual modulation of the dark atom concentration in the matter of an underground detector can explain the positive results of DAMA/NaI and DAMA/LIBRA experiments.

In essence, the dark atom hypothesis involves only one element of BSM physics—the existence of stable multiple charged particles. All the observable features of dark atoms are dominantly related to the interaction of their helium shell, which in principle should be described by SM physics. However, the lack of usual approximations of atomic physics (smallness of core relative to the size of the atomic shell as well as electroweak character of electronic interaction) moves the analysis to the unknown sector of known physics. The developed approach to the rigorous quantum mechanical description of dark atom nuclear physics should lead to a definite answer on the production of anomalous isotopes or the existence of a low-energy bound-state in dark atom interactions with nuclei. The definiteness of these answers makes the model falsifiable and thus deserving of serious attention.

Dark atoms represent the form of strongly (nuclear)-interacting dark matter. While the results of neutrino experiments play a critical role in confirming or refuting the WIMP hypothesis, the experiments on the search for anomalous isotopes of chemical elements are more significant for studying dark atom models. In particular, the matter of the DAMA/LIBRA detector should contain anomalous sodium nuclei (bound states of these nuclei with dark atoms) with anomalously high mass. Moreover, strongly interacting heavy particles should lose their energy moving through rocks, which could lead to their increased underground concentration. Another way to test our model is the search for heavy multicharged particles at the LHC. The upper limits on the mass of such particles found in this paper indicate the possibility of testing the dark atom explanation of the dark matter density in modern experiments. We hope that the development of the correct quantum-mechanical description of dark atom nuclear physics will reveal the role of this form of dark matter in primordial metallicity, O-nuclearites (neutral nuclear droplets with nuclear electric charge compensated by the corresponding number of X-particles), and anomalous multiple-charged leptonic X-components of cosmic rays [7]. All these predictions make our approach falsifiable, refuting (or confirming!) its relevance to reality.