#### 3.3. Physics and Cosmology of Hypercolor SU(4) and SU(6) Models

So, in the simplest case of zero hypercharge, it is possible to consider some experimentally observed consequences of SU(4) minimal model [

80]. As it is seen from above, even in the minimal scenario of this type of hypercolor extension, there emerges a significant number of additional degrees of freedom. These new states, such as pNG or other hyperhadrons, can be detected in reactions at the collider at sufficiently high energies.

Readers should be reminded here of several papers that concern the formulation and construction of a vector-like hypercolor scheme [

67,

68]. In addition to the awareness of the original ideology, which made it possible to avoid known difficulties of Technicolor, main potentially observable consequences of this type of the SM extension were analyzed qualitatively and quantitatively [

69,

71,

72,

78]. Also in these articles, the possibilities of the hypercolor models for explaining the nature of DM particles were discussed in detail. Namely, this extension of the SM offers several different options as DM candidates with specific features and predictions. Some of these scenarios will be discussed in more detail below.

The vector-like hypercolor model contains two different scalar states with zero (or small) mixing, the Higgs boson and

$\tilde{\sigma}$, and possibility to analyze quantitatively an effect of this mixing tends to zero. Then, we should hope to find some New Physics signals not in channels with the Higgs boson, but from production and decays of

$\tilde{\sigma}$-meson (as shown by experimental data at the LHC, almost all predictions of the SM for the Higgs boson production cross sections in different channels as well as the widths of various modes of its decay are confirmed). Interestingly, the fluke two-photon signal at

$750\phantom{\rule{0.166667em}{0ex}}\mathrm{GeV}$ at the LHC seemed to indicate unambiguously decay of a scalar analog of the Higgs. If this were the case, the hyperpion mass in this model would have to be sufficiently small

$\sim {10}^{2}\phantom{\rule{0.166667em}{0ex}}\mathrm{GeV}$ due to the direct connection between

$\tilde{\pi}$ and

$\tilde{\sigma}$ masses. The condition of small mixing of scalars

H and

$\tilde{\sigma}$ in the conformal limit is

${m}_{\tilde{\sigma}}\approx \sqrt{3}{m}_{\tilde{\pi}}$ [

68] and it means, in fact, that

$\tilde{\sigma}$ is a pNG boson of conformal symmetry. Then, it should be close in mass to other pNG states. In this case, signals of formation and decays of charged and neutral (stable!) H-pions would be observed at the collider [

70,

80]. Nature, however, turned out to be more sophisticated.

To consider the phenomenological manifestations, we postulate a certain hierarchy of scales for numerous degrees of freedom in the model. Namely, the pNG bosons are the lightest in the spectrum of possible hyperhadrons, and the triplet of H-pions are the lightest states of pNG. This arrangement of the scales follows from the assumption that the apparent violation of the symmetry SU(4) is a small perturbation by analogy with the violation of the dynamic symmetry in the orthodox QCD scheme. There, the chiral symmetry is broken on a scale much larger than the mass scale of light quarks.

In the absence of new physics data from the LHC, we can use an estimate obtained on the assumption that the stable states in the model are dark matter candidates. In particular, the neutral H-pion ${\tilde{\pi}}^{0}$ and neutral hyperbaryon, ${B}^{0}$, can be such candidates. In this case, the analysis of the relic concentration of the dark matter makes it possible to estimate the range of masses of these particles. Thus, there is a natural mutual influence and collaboration of astrophysical and collider studies. So, in this scenario of the Standard Model extension, it becomes possible to identify DM particles with two representatives of the pNG states. For quantitative analysis, however, a more accurate consideration of the mass spectrum of the H-pion triplet and mass splitting between ${\tilde{\pi}}^{0}$ and hyperbaryon ${B}^{0}$ is necessary.

As for the mass splitting in the H-pion triplet, this parameter is defined by purely electroweak contributions [

72,

98] and is as follows:

Here,

${\mu}_{V}={M}_{V}^{2}/{m}_{\tilde{\pi}}^{2},\phantom{\rule{1.em}{0ex}}{\beta}_{V}=\sqrt{1-{\mu}_{V}/4}$, and

${G}_{F}$ denotes Fermi’s constant. Taking the H-pion mass in a wide range 200–1500 GeV, from (

94) we found the value

$\Delta {m}_{\tilde{\pi}}\approx $ 0.162–0.170 GeV.

Indeed, this small, non-zero and almost constant splitting of the mass in the triplet of the hyperpions obviously violates isotopic invariance. But at the same time, HG-parity remains a conserved quantum number. The reason is that the HG-parity is associated with a discrete symmetry, and not with a continuous transformation of the H-pion states. It is important to note that the inclusion of higher order corrections cannot destabilize a neutral weakly interacting H-pion, which is the lightest state in this pseudoscalar triplet. But charged H-pion states should decay by several channels.

In the strong channel, the width of the charged H-pion decay [

80] can be written as

Here,

${f}_{\pi}=132\phantom{\rule{0.166667em}{0ex}}\mathrm{MeV}$ and

${\pi}^{\pm}$ denotes a standard pion. The reduced triangle function is defined as

For the decay in the lepton channel we get:

where

${q}_{1}^{2}={m}_{l}^{2}$,

${q}_{2}^{2}={(\Delta {m}_{\tilde{\pi}})}^{2}$, and

${m}_{l}$ is a lepton mass.

Now, we can estimate decay widths of the charged H-pion and, correspondingly, lifetimes, and track lengths in these channels. To do this, we use (

96), (

97), and

$\Delta {m}_{\tilde{\pi}}$ from (

94) and get

Then, at TeV scale, characteristic manifestations of H-pions can be observed in the Drell–Yan type reactions due to the following fingerprints:

- (1)
large ${E}_{T,mis}$ reaction due to production of stable ${\tilde{\pi}}^{0}$ and neutrino from ${\tilde{\pi}}^{\pm}$ and/or ${W}^{\pm}$ decays, or two leptons from charged H-pion and W decays (this is reaction of associated production, $W,\phantom{\rule{0.166667em}{0ex}}{\tilde{\pi}}^{\pm},\phantom{\rule{0.166667em}{0ex}}{\tilde{\pi}}^{0}$ final state of the process);

- (2)
large ${E}_{T,mis}$ due to creation of two stable ${\tilde{\pi}}^{0}$ and neutrino from decay of charged H-pions, ${\tilde{\pi}}^{\pm}$, one lepton from ${\tilde{\pi}}^{\pm}$ and two quark jets from ${W}^{\pm}$ decay (the same final state with particles $W,\phantom{\rule{0.166667em}{0ex}}{\tilde{\pi}}^{\pm},\phantom{\rule{0.166667em}{0ex}}{\tilde{\pi}}^{0}$);

- (3)
large ${E}_{T,mis}$ due to two stable neutral H-pions and neutrino from charged H-pion decay, two leptons (virtual Z, and ${\tilde{\pi}}^{+},\phantom{\rule{0.166667em}{0ex}}{\tilde{\pi}}^{-}$ in the final state);

- (4)
large ${E}_{T,mis}$ due to two final neutral H-pions and neutrino from ${\tilde{\pi}}^{\pm}$, one lepton which originated from virtual W, ${\tilde{\pi}}^{+},\phantom{\rule{0.166667em}{0ex}}{\tilde{\pi}}^{0}$ final states.

Besides, H-pion signals can be seen due to two tagged jets in vector-boson-fusion channel in addition to main characteristics of the stable hyperpion—${E}_{mis}\sim {m}_{\tilde{\pi}}$ and accompanying leptons.

Obviously, targeted search for such signals is possible only when we know, at least approximately, the range of hyperpion mass values. These estimates can be obtained by calculating the relic content of hidden mass in the Universe and comparing it with recent astrophysical data. Within the framework of the model, such calculations were made (see below). The possible values of the H-pion mass are in the range from 600–700 GeV to 1200–1400 GeV, while the naturally $\tilde{\sigma}$-meson is quite heavy—we recall that its mass is directly related to the masses of H-pions in the case of small H-$\tilde{\sigma}$ mixing. Cross sections of the reactions above (with large missed energy and momentum) are too small to be detected without special and careful analysis of specific events with predicted signature. The number of these events is also evidently small, and the signal can be hardly extracted from the background because there is a lot of events with decaying W-bosons and, correspondingly, neutrino or quark jets.

So, another interesting process to probe into the model of this type is the production and decay of a scalar H-meson $\tilde{\sigma}$ at the LHC; this production is possible at the tree level, however, the cross section is strongly damped due to small mixing. A small value of the mixing angle, ${\theta}_{s}$, suppresses the cross section by an extra multiplier ${sin}^{2}{\theta}_{s}$ in comparison with the standard Higgs boson production.

However, at one-loop level it is possible to get single and double H-sigma in the processes of vector-vector fusion. Namely, in ${V}^{*}{V}^{\prime *}\to \tilde{\sigma},\phantom{\rule{0.166667em}{0ex}}2\tilde{\sigma}$ and/or in the decay through hyperquark triangle loop $\Delta $, i.e., ${V}^{*}\to \Delta \to {V}^{\prime}\tilde{\sigma},\phantom{\rule{0.166667em}{0ex}}2\tilde{\sigma}$. Here, ${V}^{*}$ and ${V}^{\prime}$ are intermediate or final vector bosons.

Now, a heavy H-sigma can decay via loops of hyperquarks or hyperpions

$\tilde{\sigma}\to {V}_{1}{V}_{2}$, where

${V}_{1,2}=\gamma ,Z,W$. Besides, the main decay modes of H-sigma are

$\tilde{\sigma}\to {\tilde{\pi}}^{0}{\tilde{\pi}}^{0},\phantom{\rule{0.166667em}{0ex}}{\tilde{\pi}}^{+}{\tilde{\pi}}^{-}$; these are described by tree-level diagrams that predict large decay width for

${m}_{\tilde{\sigma}}\u2a7e2{m}_{\tilde{\pi}}$. As we will see below (from the DM relic abundance analysis), at some values of

$\tilde{\pi}$ and

$\tilde{\sigma}$ masses these channels are opened. In the small mixing limit, the width is

and it depends strongly on the parameter

${\lambda}_{11}$.

An initial analysis of the model parameters was carried out in [

68], using the value

${\lambda}_{11}$ (it is denoted there as

${\lambda}_{HC}$) and

u, from (

99) we get:

$\mathsf{\Gamma}(\tilde{\sigma}\to \tilde{\pi}\tilde{\pi})\gtrsim 10\phantom{\rule{0.166667em}{0ex}}\mathrm{GeV}$ when

${m}_{\tilde{\sigma}}\gtrsim 2{m}_{\tilde{\pi}}$.

The smallness of

H–

$\tilde{\sigma}$ mixing, as it is dictated by conformal approximation, results in the multiplier

${sin}^{2}{\theta}_{s}$ for all tree-level squared amplitudes for decay widths. Then, for

$\tilde{\sigma}$ decay widths we have

Here, ${m}_{f}$ is a mass of standard fermion f and ${c}_{W}=cos{\theta}_{W}$.

Recall that the two-photon decay of the Higgs boson is the very main channel in which the deviation of the experimental data from the predictions of the SM was originally found. Analogically, we consider a

$\tilde{\sigma}\to \gamma \gamma $ decay which occurs through loops of heavy hyperquarks and H-pions; the width has the following form:

Here,

${F}_{Q}$,

${F}_{\tilde{\pi}}$,

${F}_{W}$, and

${F}_{\mathrm{top}}$ are contributions from the H-quark, H-pion,

W-boson, and top-quark loops; they can be presented as follows:

and

As it is seen, contributions from

W- and

t-quark loops are induced by non-zero

$\tilde{\sigma}$−H mixing. Taking necessary parameters from [

68], the width is evaluated as

$\mathsf{\Gamma}(\tilde{\sigma}\to \gamma \gamma )\approx 5$–10 MeV.

Obviously, the process $p\overline{p}\to \tilde{\sigma}\to \mathrm{all}$ should be analyzed quantitatively after integration of cross section of quark subprocess with partonic distribution functions. It is reasonable, however, to get an approximate value of the vector boson fusion cross section $VV\to \tilde{\sigma}\left(s\right)\to \mathrm{all}$, $V=\gamma ,Z,W$.

The useful procedure to calculate the cross section with a suitable accuracy is the method of factorization [

99]; this approach is simple and for the cross section estimation it suggests a clear recipe:

where

$\tilde{\sigma}\left(s\right)$ is

$\tilde{\sigma}$ in the intermediate state having energy

$\sqrt{s}$. A partial decay width is denoted as

$\mathsf{\Gamma}(\tilde{\sigma}\left(s\right)\to VV)$. The density of probability,

${\rho}_{\tilde{\sigma}}\left(s\right),$ can be written as

Here,

${\mathsf{\Gamma}}_{\tilde{\sigma}}\left(s\right)$ is the total width of virtual

$\tilde{\sigma}$-meson having

${M}_{\tilde{\sigma}}=\sqrt{s}$. At this energy we get exclusive cross section changing the numerator in (

105), namely

${\mathsf{\Gamma}}_{\tilde{\sigma}}\phantom{\rule{0.166667em}{0ex}}\to \phantom{\rule{0.166667em}{0ex}}\mathsf{\Gamma}(\tilde{\sigma}\to {V}^{\prime}{V}^{\prime})={\mathsf{\Gamma}}_{\tilde{\sigma}}\xb7Br(\tilde{\sigma}\to {V}^{\prime}{V}^{\prime})$; for the cross section now we have

Now, when ${M}_{\tilde{\sigma}}^{2}\gg {M}_{V}^{2}$ the cross section considered is determined by the branchings of H-sigma decay and the value of ${M}_{\tilde{\sigma}}$. Note, if $2{m}_{\tilde{\pi}}>{M}_{\tilde{\sigma}}$ H-sigma dominantly decays through following channels $\tilde{\sigma}\to WW,ZZ$. In this case, we get for $\tilde{\sigma}$ a narrow peak (Γ ≲ 10–100 MeV).

As we said earlier, up to the present, there are no signals from the LHC about the existence of a heavy scalar state that mixes with the Higgs boson. Therefore, we are forced to estimate the mass of the H-sigma relying on astrophysical data on the DM concentration. Namely, we can consider H-pions as stable dark matter particles and then take into account the connection of their mass with the mass of $\tilde{\sigma}$-meson in the (almost) conformal limit.

Now, we should use the cross section which is averaged over energy resolution. As a result, the value of the cross section is reduced significantly. More exactly, for $2{m}_{\tilde{\pi}}<{M}_{\tilde{\sigma}}$ the dominant channel is $\tilde{\sigma}\to \tilde{\pi}\tilde{\pi}$ with a wide peak $(\mathsf{\Gamma}\sim 10\phantom{\rule{0.166667em}{0ex}}\mathrm{GeV})$. So, $Br(\tilde{\sigma}\to VV)$ is small and consequently the cross section of H-meson prodution is estimated as very small.

Thus, with a sufficiently heavy (with mass (2–3) TeV) second scalar meson, the main fingerprint of its emergence in the reaction is a wide peak induced by the strong decay $\tilde{\sigma}\to 2\tilde{\pi}$. It is accompanied by final states with two photons, leptons and quark jets originating from decays of $WW$, $ZZ$, and standard ${\pi}^{\pm}$. Besides, it occurs with some specific decay mode of $\tilde{\sigma}$ with two final stable ${\tilde{\pi}}^{0}$. This channel is specified by a large missed energy; charged final H-pions result in a signature with missed energy plus charged leptons.

As it was shown, existence of global U(1) hyperbaryon symmetry leads to the stability of the lightest neutral H-diquark. In the scenario considered, we suppose that charged H-diquark states decay to the neutral stable one and some other particles. Moreover, we also assume that these charged H-diquarks are sufficiently heavy, so their contributions into processes at (1–2) TeV are negligible [

80].

Thus, having two different stable states—neutral H-pion and the lightest H-diquark with conserved H-baryon number—we can study a possibility to construct dark matter from these particles. This scenario with two-component dark matter is an immanent consequence of symmetry of this type of SM extension. Certainly, emergence of a set of pNG states together with heavy hyperhadrons needs careful and detailed analysis. At first stage, the mass splitting between stable components of the DM, not only H-pions, should be considered. Importantly, the model does not contain stable H-baryon participating into electroweak interactions. It means that any constraints for the DM relic concentration are absent for this case.

Note,

${\tilde{\pi}}^{0}$ and

${B}^{0}$ have the same tree level masses, so it is important to analyze the mass splitting

$\Delta {M}_{B\tilde{\pi}}={m}_{{B}^{0}}-{m}_{{\tilde{\pi}}^{0}}$. As it follows from calculations, this parameter depends on electroweak contributions only, all other (strong) diagrams are canceled mutually. Then, we get:

where

$\beta =\sqrt{1-{\displaystyle \frac{{M}_{W}^{2}}{4{m}_{\tilde{\pi}}^{2}}}}$. An important point is that

$\Delta {M}_{B\tilde{\pi}}$ dependence on a renormalization point results from the coupling of the pNG states with H-quark currents of different structure at close but not the same energy scales. Thus, the dependence of the characteristics of the DM on the renormalization parameter is necessarily considered when analyzing the features of the DM model.

We remind that it is assumed that not-pNG H-hadrons (possible vector H-mesons, etc.) manifest itself at much more larger energies. It results from the smallness of the scale of explicit SU(4) symmetry breaking comparing with the scale of dynamical symmetry breaking. This hierarchy of scales copies the QCD construction.

Now, since the effects of hyperparticles at the collider are small, and an interesting scenario of a two-component DM arises, let us consider in more detail the possibility of describing the dark matter candidates in the framework of this model [

100,

101]. At the same time, we should note the importance of previous studies of the DM scenarios based on vector-like technicolor in the papers ref. [

65,

68,

69,

71,

72,

102,

103,

104]. Several quite optimistic versions of the DM description (including technineutrons, B-baryons, etc.) were considered, which, however, did not have a continuation, since they relied on a number of not quite reasonable assumptions—in particular, that B-baryons form the triplet.

When we turn to the hypercolor model, it will be necessary not only to calculate the total annihilation cross section for dark matter states, but to analyze the entire kinetics of freezing out of DM particles. The reason is that the mass splittings in the H-pions multiplet and between masses of two components are small (in the last case this parameter can be suggested as small). Then, the coupled system of five Boltzmann kinetic equations should be solved. Namely, two states of the neutral H-baryon,

${B}^{0},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\overline{B}}^{0}$, neutral H-pion and also two charged H-pions should be considered. Such cumbersome kinetics are a consequence of proximity of masses of all particles participating in the process of formation of residual DM relic concentration. So, the co-annihilation processes [

105] can contribute significantly to the cross section of annihilation. It had been shown also in previous vector-like scenario [

72].

Now, for each component of the DM and co-annihilating particles,

$i,j={\tilde{\pi}}^{+},\phantom{\rule{0.166667em}{0ex}}{\tilde{\pi}}^{-},{\tilde{\pi}}^{0};\mu ,\nu =B,\phantom{\rule{0.166667em}{0ex}}\overline{B}$, we have the basic Boltzmann EquationS (

108) and (

109) (neglecting reactions of type

$iX\leftrightarrow jX$):

Because of decays of charged H-pions, the main parameters in this calculation are total densities of

$\tilde{\pi}$,

${B}^{0}$ and

$\overline{{B}^{0}}$, namely,

${n}_{\tilde{\pi}}={\sum}_{i}{n}_{i}$ and

${n}_{B}={\sum}_{\mu}{n}_{\mu}$. Using an equilibrium density,

${n}_{eq}$, for describing co-annihilation, we estimate

${n}_{i}/n\approx {n}_{i}^{eq}/{n}^{eq}$. Then, the system of equations can be rewritten as

where

Now, it is reasonable to consider the ratio

${m}_{\tilde{\pi}}/{M}_{B}\approx 1$ using a suitable value of the renormalization parameter in the mass splitting between

${m}_{{\tilde{\pi}}^{0}}$ and

${M}_{{B}^{0}}$, more exactly,

$\Delta {M}_{{B}^{0}{\tilde{\pi}}^{0}}/{m}_{{\tilde{\pi}}^{0}}\lesssim 0.02$. Then, the system of kinetic equations simplifies further. Having

${n}_{B}^{eq}/{n}_{\tilde{\pi}}^{eq}=2/3$, we come to the following form of equations:

When masses of the DM components are close to each other, it is necessary to take into account the temperature dependence [

105] in the cross sections

$<\sigma v{>}_{\tilde{\pi}\tilde{\pi}}$ and

$<\sigma v{>}_{BB}$ of the processes:

where

$x={m}_{\tilde{\pi}}/T$,

v is the relative velocity of final particles.

There are commonly used notations, which are convenient for solve the system,

$Y=n/s$ and

$x={m}_{\tilde{\pi}}/T$, where

s is the density of entropy. Then, neglecting small terms

$\Delta {M}_{\tilde{\pi}}/{M}_{B}$ we have:

The energy density is determined by a set of relativistic degrees of freedom, this function can be written in a convenient form as

Here, we use an approximated value of this parameter, which works in the numerical solution with a good accuracy [

101] and better than known approximation

$g\left(T\right)\approx 100$; also, there are standard notations from [

106]:

${\lambda}_{i}=2.76\times {10}^{35}{m}_{\tilde{\pi}}<\sigma v{>}_{i}$,

${Y}_{\tilde{\pi}}^{eq}=0.145(3/g\left(T\right)){x}^{3/2}{e}^{-x},\phantom{\rule{0.166667em}{0ex}}{Y}_{B}^{eq}=0.145(2/g\left(T\right)){x}^{3/2}{e}^{-x}$,

The DM relic density,

$\mathsf{\Omega}{h}^{2}$, is expressed in terms of relic abundance and critical mass density,

$\rho $ and

${\rho}_{crit}$:

Present time values are denoted by the subscript “0”.

After the replacement

$W=logY$ [

106], the system of kinetic equations is solved numerically. As it is shown in detail in [

101], there is a set of regions in a plane of H-pion and H-sigma masses, where it is possible to get the value of the DM relic density in a correspondence with the modern astrophysical data. More exactly, the H-pion fraction is described by the following intervals:

$0.1047\le \mathsf{\Omega}{h}_{HP}^{2}+\mathsf{\Omega}{h}_{HB}^{2}\le 0.1228$ and

$\mathsf{\Omega}{h}_{HP}^{2}/(\mathsf{\Omega}{h}_{HP}^{2}+\mathsf{\Omega}{h}_{HB}^{2})\le 0.25$). There are also some slightly different areas having all parameters nearly the same. However, H-pions make up just over a quarter of dark matter density, more exactly,

$0.1047\le \mathsf{\Omega}{h}_{HP}^{2}+\mathsf{\Omega}{h}_{HB}^{2}\le 0.1228$ and

$0.25\le \mathsf{\Omega}{h}_{HP}^{2}/(\mathsf{\Omega}{h}_{HP}^{2}+\mathsf{\Omega}{h}_{HB}^{2})\le 0.4$. Certainly, there are regions of parameters which are forbidden by restrictions by XENON collaboration [

13,

107,

108].

It is important that there are no regions of parameters where the H-pion component dominates in the dark matter density. The reason is that this hyperpion component interacts with vector bosons, $Z,\phantom{\rule{0.166667em}{0ex}}W$, at the tree level and, consequently, annihilates into ordinary particles much faster than stable ${B}^{0}$-baryons. The latter particles do not interact with standard vector bosons directly but only at loop level through H-quark and H-pion loops. It is a specific feature of SU(4) vector-like model having two stable pNG states.

At this stage of analysis (without an account of loop contributions from ${B}^{0}-{B}^{-}$ annihilation), there are three allowable regions of parameters (masses):

Area 1: here ${M}_{\tilde{\sigma}}>2{m}_{{\tilde{\pi}}^{0}}$ and $u\ge {M}_{\tilde{\sigma}}$; at small mixing, ${s}_{\theta}\ll 1$, and large mass of H-pions we get a reasonable value of the relic density and a significant H-pion fraction;

Area 2: here again ${M}_{\tilde{\sigma}}>2{m}_{{\tilde{\pi}}^{0}}$ and $u\ge {M}_{\tilde{\sigma}}$ but ${m}_{\tilde{\pi}}\approx 300$–600 GeV; H-pion fraction is small here, approximately, (10–15)%;

Area 3: ${M}_{\tilde{\sigma}}<2{m}_{\tilde{\pi}}$—this region is possible for all values of parameters, but decay $\tilde{\sigma}\to \tilde{\pi}\tilde{\pi}$ is prohibited and two-photon signal from reaction $pp\to \tilde{\sigma}\to \gamma \gamma X$ would have to be visible at the LHC. Simultaneously, H-pion fraction can be sufficiently large, up to $40\%$ for large ${m}_{{\tilde{\pi}}^{0}}\sim 1\phantom{\rule{0.166667em}{0ex}}\mathrm{TeV}$ and small angle of mixing.

Thus, from kinetics of two components of hidden mass it follows that the mass of these particles can vary in the interval (600–1000) GeV in agreement with recent data on the DM relic abundance. Having these values, it is possible to consider some manifestations of the hidden mass structure in the model.

Particularly, the inelastic interactions of high-energy cosmic rays with the DM particles can be interesting for studying the hidden mass distribution using signals of energetic leptons (neutrino) or photons which are produced in this scattering process [

101].

Cosmic ray electrons can interact with the H-pion component via a weak boson in the process $e{\tilde{\pi}}^{0}\to {\nu}_{e}{\tilde{\pi}}^{-}$, then charged ${\tilde{\pi}}^{-}$ will decay. In the narrow-width approximation we get for the cross section: $\sigma (e{\tilde{\pi}}^{0}\to {\nu}_{e}{\tilde{\pi}}^{0}l{\nu}_{l}^{\prime})\approx \sigma ((e{\tilde{\pi}}^{0}\to {\nu}_{e}{\tilde{\pi}}^{-})\xb7Br({\tilde{\pi}}^{-}\to {\tilde{\pi}}^{0}l{\nu}_{l}^{\prime}),$ branchings of charged hyperpion decay channels are: $Br({\tilde{\pi}}^{-}\to {\tilde{\pi}}^{0}e{\nu}_{e}^{\prime})\approx 0.01$ and also $Br({\tilde{\pi}}^{-}\to {\tilde{\pi}}^{0}{\pi}^{-})\approx 0.99$.

Considering final charged hyperpion ${\tilde{\pi}}^{-}$ near its mass shell, standard light charged pion produces neutrino $e{\nu}_{e}$ and $\mu {\nu}_{\mu}$ with following probabilities: ≈$1.2\times {10}^{-6}$ and ≈0.999, correspondingly.

Then, in this reaction, an energetic cosmic electron produces electronic neutrino due to vertex

$We{\nu}_{e}$, and soft secondary

${e}^{\prime}{\nu}_{e}^{\prime}$ or

$\mu {\nu}_{\mu}$ arise from charged H-pion decays. Now, there are final states with

$Br\left({\tilde{\pi}}^{0}{\nu}_{e}{\mu}^{\prime}{\nu}_{\mu}^{\prime}\right)\approx 0.99$ and

$Br\left({\tilde{\pi}}^{0}{\nu}_{e}{e}^{\prime}{\nu}_{e}^{\prime}\right)\approx {10}^{-2}$. Obviously, we use here some simple estimations, they can be justified in the framework of the factorization approach [

99]. Characteristic values of H-pion mass which can be used for the analysis are, for example,

${m}_{{\tilde{\pi}}^{0}}=800\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{GeV}$ and

$1200\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{GeV}$.

As it results from calculations, at initial electron energies in the interval

${E}_{e}=$(100–1000) GeV the cross section of the process decreases from

$O\left(10\right)\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{nb}$ up to

$O\left(0.1\right)\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{nb}$ having maximum at small angles between electron and the neutrino emitted, i.e., inelastic neutrino production occurs in the forward direction (for more detail see figures in Ref. [

101]). In this approximation, the energy of the neutrino produced is proportional to the energy of the incident electron and depends on the mass of the dark matter particle very weakly. The neutrino flux is calculated by integrating of spectrum,

$dN/d{E}_{\nu}$, this flux depends on H-pion mass very weakly. In the interval (50–350) GeV it decreases most steeply, and then, down to energies ∼1 TeV the fall is smoother.

Certainly, integrating the spectrum

$dN/d{E}_{\nu}$, we can estimate the number of neutrino landing on the surface of IceTop [

109,

110] which is approximately one squared kilometer.

Even taking into account some coefficients to amplify the DM density near the galaxy center for the symmetric Einasto profile, we have found the number of such neutrino events per year as very small,

${N}_{\nu}=$ (6–7), in comparison with the corresponding number of events for neutrino with energies in the multi-TeV region. Note, the Einasto profile modified in such manner reproduces well the hidden mass density value near Galaxy center in concordance with other DM profiles [

111]. In any case, such small number of neutrino events at IceCube does not allow to study this interaction of cosmic rays with the DM effectively. Any other DM profile gives practically the same estimation of number of events for neutrino with these energies. Indeed, cross section of

$\nu N$ interaction for small neutrino energies ∼(

${10}^{2}$–

${10}^{3}$) GeV is much lower than for neutrino energies ∼(

${10}^{1}$–

${10}^{5}$) TeV. Consequently, all parameters of the signal detected, particularly, deposition of energy, intensity of Cherenkov emission, are noticeably worse. Because of the absence of good statistics of neutrino events, it is practically impossible to measure the neutrino spectrum of the predicted form. Probability of neutrino detection can be estimated in the concept of an effective area of the detector [

110,

112,

113,

114,

115]). In our case this probability is small,

$P={10}^{-10}\u2013{10}^{-8}$ [

101], so we need some additional factor that can increase the flux of neutrino substantially.

In principle, some factors amplifying these weak signals of cosmic electron scattering off the DM can be provided by inhomogeneities in the hidden mass distribution, i.e., so called clumps [

116,

117,

118,

119,

120,

121]. The scattering of cosmic rays off clusters of very high density [

122] can result in amplifying neutrino flux substantially [

123,

124].

Though the scattering process suggested can be seen due to specific form of neutrino flux, the expected number of events is too small to be measured in experiments at modern neutrino observatories. The weakeness of the signal is also resulted from effective bremsstrahlung of electrons and the smallness of electron fraction in cosmic rays, ∼1%. Therefore, they are not so good probe for the DM structure; only if there are sharply non-homogeneous spatial distribution of hidden mass, the signal of production of energetic neutrino by cosmic electrons can be detected. It is an important reason to study inelastic scattering of cosmic protons, because they are more energetic and have a much larger flux.

Besides, an important information of the nature and profile of hidden mass should be manifested in a specific form of the DM annihilation gamma spectrum from clumps [

125,

126]. This signal can be significantly amplified due to increasing of density of hidden mass inside clumps, corresponding cross section depends on the squared density in contrary with the energy spectrum of final particles (neutrino, for example) which is resulted from scattering. In the last case, cross section is proportional to a first degree of the DM density.

Indeed, in the vector-like model with the two lowest in mass neutral stable states, one from these components does not participate in the scattering reaction with leptons, so the flux of final particles is diminished. There are, however, annihilation channels of both components into charged secondaries which emit photons. Some important contributions into this process describe so called virtual internal bremsshtrahlung (VIB). This part of photon spectrum containing information on the DM structure may be about 30%. Consequently, a feature of the DM structure in the model (particularly, existence of two components with different tree-level interactions) can result in some characteristic form of the annihilation diffuse spectra.

Introducing a parameter which determines H-pion fraction in the DM density,

full annihilation spectrum is written as

Contributions of the DM components to the total cross section of production of diffuse photons differ because of distinction in tree-level interaction with weak bosons. Annihilation of hyperpions into charged states (in particular, W-bosons) gives the most intensive part of diffuse photon flux. However, ${B}^{0}$-baryons can provide a significant fraction of this flux in some regions of the model parameters, namely, if the DM particles have mass ≈600 GeV. It should be noted also that $\sigma $-meson mass affects the cross section value changing it noticeably: from $-10\%$ to $+50\%$, approximately. This effect is seen better for the DM component mass ≈800 GeV when contribution from ${B}^{0}$ is not so prominent. Obviously, contributions to the gamma flux intensity from annihilation of different DM components contribute to the gamma flux in correspondence with the model content and structure. Thus, there appears a sign of the existence of two Dark matter components, observed in the form of a specific humped curve of the photon spectrum, due to virtual internal bremsstrahlung subprocesses. This effect should be considered in detail, because it is necessary to have much more astrophysics data together with an accurate analysis of all possible contributions to the spectrum for various regions of parameters. Certainly, because of high density of interacting particles, reactions of annihilation into photons in the DM clumps can be seen much better, and it also should be studied.

There is a set of possible observing consequences of the DM particle origin, structure and interactions produced by vector-like extension of the SM. Some of them have been analyzed quantitatively, while the analysis of others is still in progress. As there are plenty of additional heavy degrees of freedom in this model, they induce new effects that should be considered to predict observable and measurable phenomena. Moreover, the numerical estimations of model parameters and analysis of the effects above are based on some assumptions about spectrum of hyperhadrons. Particularly, it is suggested that charged di-hyperquark states, ${B}^{\pm}$, have masses which are much larger than neutral-state masses. It allows us to eliminate a lot of possible subprocesses with these particles and simplify substantially the system of kinetic equations for the DM components. This approach is quite reasonable.

Extension of the vector-like model symmetry, from SU(4) to SU(6), unambiguously results in a much larger number of additional H-hadrons which spawn a great quantity of new processes and effects. As noted above, there is an invariance of the model physical Lagrangian with respect to some additional symmetries, as a result of which we obtain a number of stable states and it is necessary to study their possible manifestations. Consideration of a new variant of the vector-like model of H-quarks is at the very beginning, therefore now we can define only some possible scenarios.

In the scenario with $Y={Y}_{S}=0$, two stable neutral states, H-pion, ${B}^{0}$ and also the lightest charged ${B}^{\pm}$ occur, as it is dictated by hyper-G-parity. In this case, we again have the opportunity to construct hidden mass from several components, as it was done in the previous version of SU(4) symmetry. However, a quantitative analysis of the mass difference for B-diquarks is necessary in order to assess the importance of the co-annihilation process for them. Assuming this mass splitting to be small, one can predict that the characteristics of a two-component DM in this scenario will not differ much from the previous version. Namely, we expect the masses of all dark matter components to be in the interval (0.8–1.2) TeV providing corresponding DM density.

Very interesting consequences follow from an occurrence of the stable charged state. First, the charged H-hadrons interact electromagnetically with cosmological plasma, so the hidden mass can be split from the plasma much later in comparison with the purely neutral DM. Second, there should be tree-level annihilation of these DM states into photons with an observable flux of specific form. Certainly, these conclusions make sense if relative concentration of the charged component is not small. Known data on the gamma spectrum from cosmic telescopes should help to establish necessary restrictions for the scenario parameters.

Moreover, the stable charged H-hadron can be seen in the collider experiments at corresponding energies. These heavy particles in the final states and neutral stable particles should be observed in the characteristic events with large missed energy. The cross sections of reactions and energies of detectable secondaries (hadronic jets and/or leptons) depend on the mass splitting between neutral and charged states.

The charged stable H-hadron should also be prominent in the scattering off the nuclei in underground experiments, we can expect that the corresponding cross section will be larger then in the stable neutral component scattering due to exchanges via vector bosons, not only through intermediate scalar mesons. However, known restrictions for measurable cross sections which follow from experiments at the underground setup will predict then more heavy stable particles in this model.

Note also that the next possible scenario with ${Y}_{Q}=0$, ${Y}_{S}=\pm 1/2$ is less interesting for describing the DM properties because of the absence of stable states. (There is, possibly, a very special case with the one H-baryon state stable, the case should be considered separately, this work is in progress.).

Considering the physical Lagrangian for SU(6) vector-like extension, we find an important feature of the model: in this case there arise interactions of K-doublets and B-states with standard vector bosons (see the section above). These interactions can both amplify channels of new particles production at the collider and increase cross section of the DM annihilation and co-annihilation. Then, possible value of the DM components mass should also be larger to provide a suitable hidden mass density. In any case, these scenarios should be carefully analyzed before we can formulate a set of predictions for collider and astrophysical measurements. As it is seen, the vector-like extensions of the SM allow us to suggest some interesting scenarios with new stable heavy objects—H-hadrons—which can manifest itself both in events with large missed energy at the LHC and in astrophysical signals such as spectrum of photons and/or leptons from various sources in the Galaxy.