INTRODUCTION

Modern laser technologies [1, 2] employ active objects (nanoparticles, atomic defects, and neutrinos) in different periodic structures and metamaterials [3]. Femtosecond laser coherent spectroscopy [4] reveals behavioral features of active objects in these nonlinear systems. Experiments [5] to study neutrino oscillations have proved the existence of the neutrino rest mass (energy, 280 MeV) and the possibility of altering the type of a neutrino (transforming a µ neutrino into a τ neutrino). The use of these materials in space requires that we solve the problems of the effect relic radiation has on the key parameters of active objects in the field of low temperatures, lepton characteristics, and particle acceleration, caused by the varying intensity of radiation emitted by such objects. Problems then arise in studying the nature of the dark matter particles, dark energy [6–9], and chiral fractal structures of the Universe. A specific fundamental fractal (the Cantor random set) was used in [10], for which the Hausdorff dimension was an irrational number: \({\varphi } = {{(\sqrt 5 - 1)} \mathord{\left/ {\vphantom {{(\sqrt 5 - 1)} 2}} \right. \kern-0em} 2}.\) Our models are based on the fractional calculus theory and the fractal concept [11]. There is also the problem of describing the supernonradiative states (SNS) of different fields: gravitational, relic photon, Higgs, neutrino, and physical vacuum [12]. The aim of this work was to investigate the effect the Higgs boson has on the supernonradiative states of active objects, neutrino oscillations, and anomalies of lepton magnetic moments.

TEMPERATURES, ENERGIES, ACCELERATIONS, AND NEUTRINO OSCILLATIONS

In [13], the emission of single photons by atomic defects (emitters with an energy gap of 5.95 eV) of boron nitride nanotubes (BNNT) was observed in a wide range of temperatures, including ambient temperatures. The estimated upper boundary for temperature was around 800 K. In the low-temperature region, relic radiation can influence the key parameters of active nanoobjects and atomic defects. Using the anisotropic model [8, 9], estimates were obtained for relic radiation temperature \({{T}_{{\text{r}}}} = 2.72548\,\,{\text{K}}{\text{,}}\) relic radiation dipole anisotropy \({\delta }{{T}_{{\text{r}}}},\) and mean oscillations of the relic radiation temperature \({\delta }{{T}_{A}}.\) The expressions of the required characteristic temperatures and energies \({{{\varepsilon }}_{{\text{r}}}},\)\({{{\varepsilon }}_{{{\text{r}}A}}},\)\({\varepsilon }_{{{\text{r}}A}}^{'}\) can be written as

$$\begin{gathered} {{T}_{{\text{r}}}} = {{T}_{{{\text{r}}A}}} + T_{{{\text{r}}A}}^{'};\,\,{{T}_{{{\text{r}}A}}} = u_{{{\text{r}}A}}^{2}{{T}_{{\text{r}}}}; \\ T_{{{\text{r}}A}}^{'} = v_{{{\text{r}}A}}^{2}{{T}_{{\text{r}}}};\,\,{{T}_{{\text{r}}}} = {{a}_{T}}{{{\varepsilon }}_{{\text{r}}}}; \\ u_{{{\text{r}}A}}^{2} + {\text{v}}_{{{\text{r}}A}}^{2} = 1;\,\,1 - 2u_{{{\text{r}}A}}^{2} = {{(N + 1)} \mathord{\left/ {\vphantom {{(N + 1)} {z_{{A2}}^{'}}}} \right. \kern-0em} {z_{{A2}}^{'}}}; \\ {{T}_{{{\text{r}}A}}} = {{a}_{T}}{{{\varepsilon }}_{{{\text{r}}A}}};\,\,T_{{{\text{r}}A}}^{'} = {{a}_{T}}{\varepsilon }_{{{\text{r}}A}}^{'};\,\,{{N}_{{{\text{r}}A}}} = z_{{A2}}^{'} + N + 1. \\ \end{gathered} $$
(1)

Parameters \({{T}_{{{\text{r}}A}}} = 1.3390101\,\,{\text{K}}{\text{,}}\)\(T_{{{\text{r}}A}}^{'} = 1.3864699\,\,{\text{K}}{\text{,}}\)\({{{\varepsilon }}_{{\text{r}}}}\) = 469.58535 µeV, \({{{\varepsilon }}_{{{\text{r}}A}}} = 230.75328\,\,\mu {\text{eV}}{\text{,}}\)\({\varepsilon }_{{{\text{r}}A}}^{'} = 238.93207\,\,\mu {\text{eV}}{\text{,}}\) ordinary red shift \(z_{{A2}}^{'}\) = 1034.109294, \({{N}_{{{\text{r}}A}}} = 1052.116604,\) and the maximum number of active effective particles \(N = 17.0073101\) [9]. Phase transition temperatures \(T_{A}^{'},\)\({{T}_{A}}\) [9] are determined using the numbers of quanta \({{n}_{{zA}}}\) (for the ordinary red shift) and \({{n}_{{z{\mu }}}}\) (for the cosmological red shift \(z_{{\mu }}^{'}\)), and the total number of relic radiation quanta \({{N}_{{{\text{r}}a}}}\), according to the formulas

$$\begin{gathered} T_{A}^{'} = {{T}_{A}} + {\delta }{{T}_{A}};\,\,{{T}_{A}} = {{n}_{{zA}}}T_{A}^{'};\,\,{\delta }{{T}_{A}} = {{n}_{{z{\mu }}}}T_{A}^{'}; \\ {{n}_{{z{\mu }}}} = {{z_{{\mu }}^{'}} \mathord{\left/ {\vphantom {{z_{{\mu }}^{'}} {{{N}_{{{\text{r}}a}}}}}} \right. \kern-0em} {{{N}_{{{\text{r}}a}}}}};\,\,{{n}_{{zA}}} + {{n}_{{z{\mu }}}} = 1;\,\,{{N}_{{{\text{r}}a}}} = z_{{A2}}^{'} + z_{{\mu }}^{'}; \\ {\delta }{{T}_{{\text{r}}}} = {{{{Q}_{{H3}}}T_{{{\text{r}}A}}^{'}{\delta }{{T}_{A}}} \mathord{\left/ {\vphantom {{{{Q}_{{H3}}}T_{{{\text{r}}A}}^{'}{\delta }{{T}_{A}}} {T_{A}^{'}}}} \right. \kern-0em} {T_{A}^{'}}};\,\,2T_{{{\text{r}}A}}^{'} = {{N}_{{{\text{r}}A}}}T_{A}^{'}. \\ \end{gathered} $$
(2)

The parameters are \(T_{A}^{'} = 2.6355822\,\,{\text{mK}}{\text{,}}\)\({{T}_{A}} = 2.6173985\,\,{\text{mK}}{\text{,}}\)\({\delta }{{T}_{A}}\) = 18.183633 µK, \({\delta }{{T}_{{\text{r}}}} = 6.7035181\,\,{\text{mK}}{\text{,}}\) and \({{Q}_{{H3}}}\, = 0.700790572.\)

The expression for the ratio of acceleration can be written for cosmological fractal objects using the Dicke superradiation model [9]:

$${{{{g}_{0}}} \mathord{\left/ {\vphantom {{{{g}_{0}}} {{{g}_{{{\text{SE}}}}}}}} \right. \kern-0em} {{{g}_{{{\text{SE}}}}}}} = {{{{n}_{{\text{G}}}}\left( {z_{{A2}}^{'} - z_{{\mu }}^{'} + {{I}_{{\text{m}}}}I\,_{0}^{{ - 1}}} \right)} \mathord{\left/ {\vphantom {{{{n}_{{\text{G}}}}\left( {z_{{A2}}^{'} - z_{{\mu }}^{'} + {{I}_{{\text{m}}}}I\,_{0}^{{ - 1}}} \right)} 2}} \right. \kern-0em} 2}.$$
(3)

Here, \({{g}_{0}} = 980.665\,\,{\text{cm}}\,\,{{{\text{s}}}^{{ - 2}}}\) is the acceleration of gravity on the Earth’s surface; \({{g}_{{{\text{SE}}}}}\) is the Earth’s acceleration toward the Sun; and the ratio between maximum \({{I}_{{\text{m}}}}\) and initial \({{I}_{0}}\) intensities of radiation is \({{I}_{{\text{m}}}}I_{0}^{{ - 1}}\) = \(81.06580421.\) From (3), we find that \({{g}_{{{\text{SE}}}}}\) = \(0.590056\,\,{\text{cm}}\,\,{{{\text{s}}}^{{ - 2}}}.\) In light of parameters \({{n}_{{A0}}} = 58.04663887\) (for black holes), \({{N}_{{{\text{HG}}}}}\) = \(1.031830522 \times {{10}^{{16}}}\) from [9, 12], and based on

$${{g}_{{{\text{SE}}}}}{{N}_{{{\text{HG}}}}} = {{g}_{{{\text{nS}}}}}{{n}_{{A0}}}$$
(4)

we estimate the acceleration of gravity: \({{g}_{{{\text{nS}}}}}\) = \(1.0488769 \times {{10}^{{12}}}\,\,{\text{m}}\,\,{{{\text{s}}}^{{ - 2}}}\) on the surface of a neutron star, which agrees with estimate \({{10}^{{12}}}\,\,{\text{m}}\,\,{{{\text{s}}}^{{ - 2}}}\) from [14]. Note that formulas (3) and (4) allow us to generalize the description of the ratios of acceleration of active nanoobjects. Femtosecond laser coherent spectroscopy enables us to monitor variations in parameter \({{I}_{{\text{m}}}}I\,_{0}^{{ - 1}}.\)

Using the anisotropic model [8, 9], we obtain the expression of the susceptibility tensor \({{{\hat {\chi }}}_{{{\text{ef}}}}}.\) The effect \({{{\hat {\chi }}}_{{{\text{ef}}}}}\) has on characteristic energy \({{E}_{{H{\nu }}}} = 1627.379629\,\,{\text{meV}}\) determines energy tensor \({{{\hat {\varepsilon }}}_{{H{\nu }}}} = {{{\hat {\chi }}}_{{{\text{ef}}}}}{{E}_{{H{\nu }}}}\) with components \({{{\varepsilon }}_{{ij}}} = {{{\chi }}_{{ij}}}{{E}_{{H{\nu }}}}\) (\(i,j = 1,2,3\)). This allows us to estimate the neutrino rest energy: \({{{\varepsilon }}_{{{\text{HG}}}}} = {{\left| {{{{\varepsilon }}_{{21}}}\,{{{\varepsilon }}_{{12}}}} \right|}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}}\) = \(280.0460475\,\,{\text{meV}}{\text{.}}\) Using the model, \({\text{we}}\) estimate the energies that describe neutrino oscillations:

$$\begin{gathered} {{{\varepsilon }}_{{{\tau L}}}} = {{{\varepsilon }}_{{{\text{HG}}}}} + {{{\varepsilon }}_{{{\tau G}}}};\,\,\,{{{\varepsilon }}_{{{\tau G}}}} = {{{\varepsilon }}_{{2u{\tau }}}} + {{{\varepsilon }}_{{{\text{hL}}}}}; \\ {{{\varepsilon }}_{{2u{\tau }}}} = {{z}_{{2u{\tau }}}}{{{\varepsilon }}_{{{\text{HG}}}}};\,\,\,{{{\varepsilon }}_{{{\text{hL}}}}} = {{\Omega }_{{{\text{hL}}}}}{{{\varepsilon }}_{{{\text{HG}}}}}; \\ \end{gathered} $$
(5)
$$\begin{gathered} {\varepsilon }_{{{\tau L}}}^{2} - {\varepsilon }_{{{\text{HG}}}}^{2} = {{z}_{{{\tau G}}}}({{z}_{{{\tau G}}}} + 2){\varepsilon }_{{{\text{HG}}}}^{2}; \\ {{z}_{{{\tau G}}}} = {{z}_{{2u{\tau }}}} + {{\Omega }_{{{\text{hL}}}}};\,\,\,{{\Omega }_{{{\text{hL}}}}} = {{{{n}_{{{\text{h}}2}}}{{E}_{{\text{e}}}}} \mathord{\left/ {\vphantom {{{{n}_{{{\text{h}}2}}}{{E}_{{\text{e}}}}} {{{E}_{{{\text{H}}0}}}}}} \right. \kern-0em} {{{E}_{{{\text{H}}0}}}}}; \\ \end{gathered} $$
(6)
$$\begin{gathered} {{n}_{{{\text{F}{\tau }}}}} + n_{{{\text{F}{\tau }}}}^{'} = 1;\,\,\,n_{{{\text{F}{\tau }}}}^{'} = \Omega _{{{{\tau}\text{ L}}}}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}; \\ {{\Omega }_{z}} = {{\Omega }_{{z0}}} + {{\Omega }_{{{\text{hL}}}}} = 0.5 + \Omega _{{{\text{c}}2}}^{'} + {{\Omega }_{{0{{\nu }}}}} + {{\Omega }_{{{{\tau}\text{ L}}}}}. \\ \end{gathered} $$
(7)

Based on energies \({{E}_{{z0}}} = 91.188\;{\text{GeV}}\) of the \(z0\) boson and \({{E}_{{\text{e}}}} = 0.51099907\,\,{\text{MeV}}\) of an electron [15]; Higgs boson energy \({{E}_{{{\text{H}}0}}} = 125.03238\,\,{\text{GeV}}{\text{,}}\) the number of quanta of a second black hole \({{n}_{{{\text{h2}}}}} = 29.02331944\) before its merger with the first black hole \({{n}_{{{\text{h}}1}}};\) density \(\Omega _{{{\text{c}}2}}^{'} = 0.224091707\) of cold dark matter in neutron stars and density \({{\Omega }_{{0\nu }}} = 0.002939801\) of a neutrino [8, 9], we obtain \({{\Omega }_{{z0}}} = {{{{E}_{{z0}}}} \mathord{\left/ {\vphantom {{{{E}_{{z0}}}} {{{E}_{{{\text{H}}0}}}}}} \right. \kern-0em} {{{E}_{{{\text{H}}0}}}}}\) = \(0.729315078,\) ΩhL = \(1.186165 \times {{10}^{{ - 4}}},\)\({{\Omega }_{z}} = 0.729433695,\)\({{\Omega }_{{{\tau L}}}} = 0.002402187,\)\(n_{{{\text{F}{\tau }}}}^{'} = \)\(0.049012111,\)\({{n}_{{{\text{F}{\tau }}}}} = 0.950987889.\) In light of spectral parameter \({{S}_{{2u}}} = 0.033051284\) and expressions (5) and (6), we find parameters \({{z}_{{2u{{\tau }}}}} = 0.5{{n}_{{{\text{F}{\tau }}}}}{{S}_{{2u}}}\) = \(0.015715686,\) and \({{z}_{{{\tau G}}}} = 0.015834303,\) which allow a neutrino field interpretation of the cosmological red shift. The energies are \({{{\varepsilon }}_{{{\text{hL}}}}} = 33.218082\,\,\mu {\text{eV}}{\text{,}}\)\({{{\varepsilon }}_{{2u{\tau }}}} = \,4.401115748\,\,{\text{meV}}{\text{,}}\)\({{{\varepsilon }}_{{{\tau G}}}} = \,4.43433383\,\,{\text{meV}}{\text{,}}\) and \({{{\varepsilon }}_{{{\tau L}}}} = 284.4803813\,\,{\text{meV}}{\text{.}}\) The difference between squared energies \({\varepsilon }_{{{\tau L}}}^{2} - {\varepsilon }_{{{\text{HG}}}}^{2}\) = \(2503.298642\,\,{{({\text{meV}})}^{2}}\) and the value of \({{\Omega }_{{{\tau L}}}}\) virtually coincide with the difference between squared energies \(2500\,\,{{({\text{meV}})}^{2}}\) and lepton number 0.0024 from the experiment with neutrino oscillations in [5].

SUPERNONRADIATIVE STATES

Dynamic models allow us to describe not only superradiation but SNSes as well. We can write intensity \(J(t)\) of radiation [9] as

$$J(t) = {{J}_{0}}({{a}_{0}} + {{a}_{m}})[({{a}_{0}} - {{a}_{m}}) + 1].$$
(8)

Here, \({{J}_{0}}\) is the initial intensity of radiation; \({{a}_{0}}(t),\) and \({{a}_{m}}(t)\) generally depend on time \(t\) and other parameters. SNSes are states with \(J(t) = 0.\) These states can be achieved during the evolution of different transition effects (induction, avalanche, echo, and self-induced transparence). The possibility of describing them within the two models follows from (8).

Model А0

In this model, we assume that \({{a}_{0}} = - {{a}_{m}},\) where

$$\begin{gathered} {{a}_{m}} = {{\left( {z_{{A2}}^{'}} \right)}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}};\,\,\,\,a_{0}^{2} = a_{m}^{2} + {{z_{{\mu }}^{'}\left( {z_{{\mu }}^{'} + 2} \right)} \mathord{\left/ {\vphantom {{z_{{\mu }}^{'}\left( {z_{{\mu }}^{'} + 2} \right)} 4}} \right. \kern-0em} 4}{\kern 1pt} ; \\ a_{m}^{2} = z_{{A2}}^{'};\,\,{{N}_{{{\text{r}}a}}} = z_{{A2}}^{'} + z_{{\mu }}^{'}. \\ \end{gathered} $$
(9)

Two variants follow from (9). Variant В1 is when \(z_{{\mu }}^{'} = 0,\) and \({{N}_{{{\text{r}}a}}} = z_{{A2}}^{'};\) variant В2 is when \(z_{{\mu }}^{'} = - 2,\) and \({{N}_{{{\text{r}}a}}} = z_{{A2}}^{'} - 2.\) We introduce row vectors \({{\hat {N}}_{{d1}}}\) and \({{\hat {N}}_{{d2}}}\) and column vectors \(\hat {N}_{{d1}}^{ + }\) and \(\hat {N}_{{d2}}^{ + },\) respectively, for variants В1 and В2

$$\begin{gathered} {{{\hat {N}}}_{{d1}}} = \left( {{{N}_{{ra}}},\,z_{{A2}}^{'},\,z_{{\mu }}^{'}} \right) = \left( {z_{{A2}}^{'},\,z_{{A2}}^{'},\,0} \right); \\ {{{\hat {N}}}_{{d2}}} = \left( {z_{{A2}}^{'} - 2{\kern 1pt} ,\,z_{{A2}}^{'},\, - 2} \right). \\ \end{gathered} $$
(10)

We find norms \(\left| {{{N}_{{d1}}}} \right|\,,\)\(\left| {{{N}_{{d2}}}} \right|\) and angle \({{{\theta }}_{{d12}}}\) as functions of argument \(z_{{A2}}^{'}\):

$$\begin{gathered} {{{\hat {N}}}_{{d1}}}\hat {N}_{{d1}}^{ + } = 2{{\left( {z_{{A2}}^{'}} \right)}^{2}} = {{\left| {{{N}_{{d1}}}} \right|}^{2}}; \\ {{{\hat {N}}}_{{d2}}}\hat {N}_{{d2}}^{ + } = 8 + 2z_{{A2}}^{'}\left( {z_{{A2}}^{'} - 2} \right) = {{\left| {{{N}_{{d2}}}} \right|}^{2}}; \\ {{{\hat {N}}}_{{d1}}}\hat {N}_{{d2}}^{ + } = {{{\hat {N}}}_{{d2}}}\hat {N}_{{d1}}^{ + } = 2z_{{A2}}^{'}\left( {z_{{A2}}^{'} - 1} \right); \\ \cos {{{\theta }}_{{d12}}} = {{{\hat {N}}}_{{d1}}}\hat {N}_{{d2}}^{ + }{{\left| {{{N}_{{d1}}}} \right|}^{{ - 1}}}{{\left| {{{N}_{{d2}}}} \right|}^{{ - 1}}} \\ = \sqrt 2 \,\left( {z_{{A2}}^{'} - 1} \right){{\left[ {8 + 2z_{{A2}}^{'}\left( {z_{{A2}}^{'} - 2} \right)} \right]}^{{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-0em} 2}}}}. \\ \end{gathered} $$
(11)

It follows from (11) that we can change the sign of \(\cos {{{\theta }}_{{d12}}}\), depending on \(z_{{A2}}^{'}{\text{:}}\) at \(z_{{A2}}^{'} = 0,1,2\), we have \(\cos {{{\theta }}_{{d12}}} = - 0.5,0,0.5,\) respectively. We then introduce the Fermi-type density distribution functions \({{n}_{{d12}}}\) and \(n_{{d12}}^{'}\):

$$\begin{gathered} n_{{d12}}^{'} + {{n}_{{d12}}} = 1;\,\,\,\,n_{{d12}}^{'} = {{\cos }^{2}}{{{\theta }}_{{d12}}} \\ = 2{{\left( {z_{{A2}}^{'} - 1} \right)}^{2}}{{\left[ {6 + 2{{{\left( {z_{{A2}}^{'} - 1} \right)}}^{2}}} \right]}^{{ - 1}}}; \\ {{n}_{{d12}}}\, = {{\sin }^{2}}{{{\theta }}_{{d12}}} = 6\,{{\left[ {6 + 2{{{\left( {z_{{A2}}^{'} - 1} \right)}}^{2}}} \right]}^{{ - 1}}}; \\ {{B}_{{d12}}} = n_{{d12}}^{'} - {{n}_{{d12}}} \\ = {{\left[ {{{{\left( {z_{{A2}}^{'} - 1} \right)}}^{2}} - 3} \right]} \mathord{\left/ {\vphantom {{\left[ {{{{\left( {z_{{A2}}^{'} - 1} \right)}}^{2}} - 3} \right]} {\left[ {{{{\left( {z_{{A2}}^{'} - 1} \right)}}^{2}} + 3} \right]}}} \right. \kern-0em} {\left[ {{{{\left( {z_{{A2}}^{'} - 1} \right)}}^{2}} + 3} \right]}}. \\ \end{gathered} $$
(12)

We can interpret parameter \({{B}_{{d12}}}\) from (12) as the difference between the populations of states (10) and (11). The state with \({{B}_{{d12}}} = 0\) is observed at either \(z_{{A2}}^{'} = 1 + \sqrt 3 \) or \(z_{{A2}}^{'} = 1 - \sqrt 3 .\) Here, \({{\cos }^{2}}{{{\theta }}_{{d12}}}\) = \({{\sin }^{2}}{{{\theta }}_{{d12}}} = {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2},\) indicating there is a transverse component of the effective vector from (10) and (11). This allows us to apply the interpretation in terms of SNSes with possible chirality (polarization) of the structures from (10) formed by \(z_{{A2}}^{'}.\) We introduce functions of the Bose-type distribution density of \({{N}_{{zA}}},\)\(N_{{zA}}^{'}\) and \({{n}_{{zA}}},\)\(n_{{zA}}^{'}\):

$$\begin{gathered} N_{{zA}}^{'} - {{N}_{{zA}}} = 1;\,\,\,\,{{N}_{{zA}}} = {{{{{\left( {z_{{A2}}^{'} - 1} \right)}}^{2}}} \mathord{\left/ {\vphantom {{{{{\left( {z_{{A2}}^{'} - 1} \right)}}^{2}}} 3}} \right. \kern-0em} 3}; \\ n_{{zA}}^{'} - {{n}_{{zA}}} = 1;\,\,\,\,{{n}_{{zA}}} = N_{{zA}}^{{ - 1}}. \\ \end{gathered} $$
(13)

Based on (13) and \(z_{{A2}}^{'} = 1034.109294\), we find the number of bosons in the equilibrium state: \({{N}_{{0A}}} = {{N}_{{zA}}}\left( {z_{{A2}}^{'}} \right)\) = \(3.5577160 \times {{10}^{5}}.\) This allows us to introduce typical energy \({{E}_{{0A}}} = {{N}_{{0A}}}{{E}_{G}}\) = \(4.3110733\,\,{\text{eV}}\) of the gravitational field and write energy spectrum \({{E}_{{0Ai}}} = 2{{E}_{{0A}}}{{S}_{{iu}}}\) (\(i = 1,2,3,4\)). The energies of the spectrum’s branches are: acoustical, \({{E}_{{0A1}}} = 403.01271\,\,{\text{meV}}{\text{,}}\) and \({{E}_{{0A2}}} = 284.97302\,\,{\text{meV;}}\) optical, \({{E}_{{0A3}}} = 3.9080606\,\,{\text{eV}}{\text{,}}\) and \({{E}_{{0A4}}} = 4.5960463\,\,{\text{eV}}{\text{.}}\) Energy \({{E}_{{0A}}}\) is close to threshold energy \(4.3\,\,{\text{eV}},\) as was observed at neutrino-less double \({\beta }\) decay in experiments with the 136Xe isotope [15]. This allows us to interpret this energy as the Majorana neutrino rest energy.

Based on \({{E}_{{0A}}},\) and spectra \({{E}_{{{\text{HS}}i}}} = {{E}_{{{\text{H}}0}}}S_{{0i}}^{'},\) and \({{E}_{{{\text{H}}iu}}} = {{E}_{{{\text{H}}0}}}{{S}_{{iu}}}\) we find energies \({{E}_{{d{\nu }}}} = {{E}_{{{\text{H}}0}}}\left( {1 + S_{{02}}^{'}} \right)\) = \({{{{N}_{{0{\text{H}{\nu }}}}}{{E}_{{0A}}}} \mathord{\left/ {\vphantom {{{{N}_{{0{\text{H}{\nu }}}}}{{E}_{{0A}}}} {2}}} \right. \kern-0em} {2}}\) = 129.29473 GeV and \({{E}_{{{\text{HS}}1}}} = 4.94394060\,\,{\text{GeV}}{\text{,}}\)\({{{\varepsilon }}_{{du{\nu }}}}\) = \({{{{E}_{{{\text{H}}2u}}}} \mathord{\left/ {\vphantom {{{{E}_{{{\text{H}}2u}}}} {{{Q}_{{{\text{H}}6}}}}}} \right. \kern-0em} {{{Q}_{{{\text{H}}6}}}}}\) = 2.6873617 GeV, where \({{Q}_{{{\text{H}}6}}}\) = 1.537746366, and estimate the total number of neutrinos: \({{N}_{{0{\text{H}{\nu }}}}}\) = \(5.9982619 \times {{10}^{{10}}}.\) Energy \({{E}_{{d{\nu }}}}\) is close to the local minimum between the local maxima on the dependences of the number of photons on their energy, determined by detectors in the LHC experiments. Energies \({{E}_{{{\text{HS}}1}}},\)\({{E}_{{{\text{HS}}2}}},\)\({{E}_{{{\text{H}}1u}}},\)\({{E}_{{{\text{H}}2u}}},\) and \({{{\varepsilon }}_{{du{\nu }}}}\) contain information on the transmitted momenta, and energy interval (\({{{\varepsilon }}_{{du{\nu }}}},\)\({{E}_{{{\text{HS}}1}}}\)) can describe strong interactions (hadrons), and condensed baryon matter [7].

Below, we estimate the energies of the acoustical branch of dark relic photons (virtual relic photons in the condensate), \({\varepsilon }_{{dm2}}^{'} = {{N}_{{0A}}}{{{\varepsilon }}_{{\text{r}}}}\) = \(2{\varepsilon }_{{dm}}^{'}{{S}_{{2u}}}\) = \(167.07461\,\,{\text{eV}}{\text{,}}\) and dark matter, \({\varepsilon }_{{dm}}^{'} = 2.5275057\,\,{\text{keV}}{\text{.}}\) Between energies \({\varepsilon }_{{dmi}}^{'} = 2{\varepsilon }_{{dm}}^{'}{{S}_{{iu}}}\) and \({{{\varepsilon }}_{{dmi}}} = 2{{{\varepsilon }}_{{dm}}}{{S}_{{iu}}}\) of the spectrum branches, the following conditions are satisfied for dark matter with \({\varepsilon }_{{dm}}^{'}\) and \({{{\varepsilon }}_{{dm}}}\) = \(1.7872164\,\,{\text{keV}}\):

$$\begin{gathered} {\varepsilon }_{{dm}}^{'} = \;{\varepsilon }_{{dm4}}^{'} - {\varepsilon }_{{dm2}}^{'} = {\varepsilon }_{{dm3}}^{'} + {\varepsilon }_{{dm1}}^{'}; \\ {{{\varepsilon }}_{{dm}}} = \;{{{\varepsilon }}_{{dm4}}} - {{{\varepsilon }}_{{dm2}}} = {{{\varepsilon }}_{{dm3}}} + {{{\varepsilon }}_{{dm1}}}. \\ \end{gathered} $$
(14)

The energies of the spectrum branches are: acoustical, \({\varepsilon }_{{dm1}}^{'} = 236.27919\,\,{\text{eV}}{\text{,}}\)\({{{\varepsilon }}_{{dm1}}} = 167.07461\,\,{\text{eV}}{\text{,}}\) and \({{{\varepsilon }}_{{dm2}}} = 118.13959\,\,{\text{eV;}}\) optical, \({\varepsilon }_{{dm3}}^{'} = 2.2912264\,\,{\text{keV}}{\text{,}}\)\({\varepsilon }_{{dm4}}^{'} = 2.6945803\,\,{\text{keV}}{\text{,}}\)\({{{\varepsilon }}_{{dm3}}} = 1.6201418\,\,{\text{keV}}{\text{,}}\) and \({{{\varepsilon }}_{{dm4}}} = 1.9053560\,\,{\text{keV}}{\text{.}}\) Direct experiments (the DAMA/LIBRA, CoGeNT, and CRESST-II collaborations) [6] to observe the spectrum and angular distribution of γ radiation along with the modulation spectrum from the Galactic center revealed a major local maximum near 2.4 keV and two major local minima near 1.9 and 2.7 keV against the background of stochastic behavior.

In this model, energies \({{{\varepsilon }}_{{dm4}}}\) and \({\varepsilon }_{{dm4}}^{'}\) correspond to local minima (potential wells) of the optical energy branches of dark matter with rest energies \({{{\varepsilon }}_{{dm}}},\)\({\varepsilon }_{{dm}}^{'}\) and virtually coincide with the positions of local minima in the modulation spectrum. The energy \({\varepsilon }_{{dm3}}^{'} + {{{\varepsilon }}_{{dm2}}}\) = \(2.4093661\,\,{\text{keV}}\) almost coincides with the major local maximum position. Energy \({{{\varepsilon }}_{{dm2}}}\) is positioned on the acoustical branch of the spectrum and contains information about the transmitted momentum of \({\gamma }\) radiation. Information about the presence of SNSes, dark matter, and dark relic photons can therefore be detected by the local minima in the experimental \({\gamma }\) radiation spectra against the background of the stochastic signal behavior. Information can also be obtained about the SNSes of the gravitational field, detected via the presence of local minima with energies \({{E}_{{0A3}}}\) and \({{E}_{{0A4}}}\) (strain fields) in the optical branches; transmitted momenta of the gravitational field with energies \({{E}_{{0A1}}},\)\({{E}_{{0A2}}}\) (stress fields) in the acoustical branches; and rest energy \({{E}_{{0A}}}\).

Model А1

In this model, we assume that \({{a}_{0}} = {{a}_{m}} - 1.\) We introduce distribution density functions \({{a}_{m}}\) and \(a_{m}^{'}\) for Fermi-type particles and \({{N}_{{zg}}},\)\(N_{{zg}}^{'},\)\({{n}_{{zg}}},\) and \(n_{{zg}}^{'}\) for Bose-type particles:

$$\begin{gathered} {{a}_{m}} + a_{m}^{'} = 1;\,\,2{{a}_{m}} = 1 - {{b}_{m}};\,\,2a_{m}^{'} = 1 + {{b}_{m}}; \\ 4{{b}_{m}} = z_{{\mu }}^{'}\left( {z_{{\mu }}^{'} + 2} \right);\,\,z_{{A2}}^{'} = {{N}_{{{\text{r}}a}}} - z_{{\mu }}^{'}; \\ \end{gathered} $$
(15)
$$\begin{gathered} N_{{zg}}^{'} - {{N}_{{zg}}} = 1;\,\,{{N}_{{zg}}} = {{(1 + {{b}_{m}})} \mathord{\left/ {\vphantom {{(1 + {{b}_{m}})} {(1 - {{b}_{m}})}}} \right. \kern-0em} {(1 - {{b}_{m}})}}; \\ N_{{zg}}^{'} = {8 \mathord{\left/ {\vphantom {8 {\left[ {4 - z_{{\mu }}^{'}\left( {z_{{\mu }}^{'} + 2} \right)} \right]]}}} \right. \kern-0em} {\left[ {4 - z_{{\mu }}^{'}\left( {z_{{\mu }}^{'} + 2} \right)} \right]]}}; \\ \end{gathered} $$
(16)
$$\begin{gathered} n_{{zg}}^{'} - {{n}_{{zg}}} = 1;\,\,{{n}_{{zg}}} = {{(1 - {{b}_{m}})} \mathord{\left/ {\vphantom {{(1 - {{b}_{m}})} {(1 + {{b}_{m}})}}} \right. \kern-0em} {(1 + {{b}_{m}})}}; \\ n_{{zg}}^{'} = {8 \mathord{\left/ {\vphantom {8 {\left[ {4 + z_{{\mu }}^{'}\left( {z_{{\mu }}^{'} + 2} \right)} \right]}}} \right. \kern-0em} {\left[ {4 + z_{{\mu }}^{'}\left( {z_{{\mu }}^{'} + 2} \right)} \right]}}. \\ \end{gathered} $$
(17)

Parameter \({{b}_{m}} = a_{m}^{'} - {{a}_{m}}\) from (15), which allows the interpretation of the difference between populations for Fermi-type particles, confirms there is a superstate related to cosmological red shift \(z_{{\mu }}^{'}.\) On the other hand, occupational numbers \(N_{{zg}}^{'}\) and \(n_{{zg}}^{'}\) from (16) and (17) confirm the possibility of describing the SNSes of dark matter using a gluon field (\({{n}_{g}} = 8\)) renormalized by contributions due to \(z_{{\mu }}^{'}.\) Expressions (15)–(17) are subject to the condition \( - 1 \leqslant {{b}_{m}} \leqslant 1.\) When \(z_{{\mu }}^{'}\) grows, parameter \({{b}_{m}}\) exceeds 1. Functions (15)–(17) transform into new distribution density functions for Bose-type particles: \(a_{m}^{'} - \left| {{{a}_{m}}} \right| = 1,\)\(\left| {{{N}_{{zg}}}} \right| - \left| {N_{{zg}}^{'}} \right| = 1.\) For Fermi-type particles, \(n_{{zg}}^{'} + {\kern 1pt} {\kern 1pt} \left| {{{n}_{{zg}}}} \right| = 1,\) respectively. When \(z_{{\mu }}^{'} = 7.18418108\), we obtain numerical values \(\left| {{{a}_{m}}} \right| = 7.7476025,\)\(\left| {N_{{zg}}^{'}} \right| = 0.1290729,\)\(\left| {{{n}_{{zg}}}} \right| = \)\(0.885682963,\) and \(n_{{zg}}^{'} = 0.1143170.\) This allows us to determine frequency \({\nu }_{{zg}}^{'} = n_{{zg}}^{'}{{{\nu }}_{{{\text{G}}0}}}\) = \(335.00053\,\,{\text{MHz}}{\text{,}}\) close to that of \(330\,\,{\text{MHz}}\), at which dark matter dominates, according to radiofilament observations [6].

ANOMALIES OF LEPTON MAGNETIC MOMENTS

Relic radiation can induce effects of the renormalization of initial parameters: fine structure constant \({{{\alpha }}_{0}},\) electron charge \(e,\) top speed of photon propagation in vacuum \({{c}_{0}};\) rest masses \({{m}_{{\text{e}}}},\)\({{m}_{{\mu }}},\) and \({{m}_{{\tau }}}\), and magnetons \({{{\mu }}_{{\text{B}}}},\)\({{{\mu }}_{{\mu }}},\) and \({{{\mu }}_{{\tau }}}\) for electrons, muons, and \({\tau }\) leptons, respectively:

$$\begin{gathered} \hbar {{c}_{0}} = {{e}^{2}}{{{\alpha }}_{0}};\,\,{{{\mu }}_{{\text{B}}}} = {{e\hbar } \mathord{\left/ {\vphantom {{e\hbar } {2{{m}_{{\text{e}}}}}}} \right. \kern-0em} {2{{m}_{{\text{e}}}}}};\,\,{{{\mu }}_{{\mu }}} = {{e\hbar } \mathord{\left/ {\vphantom {{e\hbar } {2{{m}_{{\mu }}}}}} \right. \kern-0em} {2{{m}_{{\mu }}}}}; \\ {{{\mu }}_{{\tau }}} = {{e\hbar } \mathord{\left/ {\vphantom {{e\hbar } {2{{m}_{{\tau }}}}}} \right. \kern-0em} {2{{m}_{{\tau }}}}}. \\ \end{gathered} $$
(18)

Here, lepton magnetic moments \(\left\langle {{{{{\hat {\mu }}}}_{{\text{e}}}}} \right\rangle ,\)\(\left\langle {{{{{\hat {\mu }}}}_{{\mu }}}} \right\rangle ,\) and \(\left\langle {{{{{\hat {\mu }}}}_{{\tau }}}} \right\rangle \) for electrons, muons, and \({\tau }\) leptons, respectively, are determined by the expressions

$$\begin{gathered} 2\left\langle {{{{{\hat {\mu }}}}_{{\text{e}}}}} \right\rangle = (2 + {{\Omega }_{{{\mu e}}}}){{{\mu }}_{{\text{B}}}};\,\,2\left\langle {{{{{\hat {\mu }}}}_{{\mu }}}} \right\rangle = (2 + {{\Omega }_{{{\mu \mu }}}}){{{\mu }}_{{\mu }}}; \\ 2\left\langle {{{{{\hat {\mu }}}}_{{\tau }}}} \right\rangle = (2 + {{\Omega }_{{{\mu \tau }}}}){{{\mu }}_{{\tau }}}. \\ \end{gathered} $$
(19)

Anomalous contributions to the magnetic moments and renormalization effects are described by parameters \({{\Omega }_{{{\mu e}}}},\)\({{\Omega }_{{{\mu \mu }}}},\) and \({{\Omega }_{{{\mu \tau }}}}\) for electrons, muons, and \({\tau }\) leptons, respectively, based on leptonic number \({{\Omega }_{{{\tau }L}}}\):

$$\begin{gathered} {{\Omega }_{{{\mu e}}}} = {{\Omega }_{{{\tau L}}}} - {{\Omega }_{{{\text{HL}}}}};\,\,{{\Omega }_{{{\text{HL}}}}} = {{{{E}_{{{\text{HL}}}}}} \mathord{\left/ {\vphantom {{{{E}_{{{\text{HL}}}}}} {{{E}_{{{\text{H}}0}}}}}} \right. \kern-0em} {{{E}_{{{\text{H}}0}}}}}; \\ {{E}_{{{\text{HL}}}}} = n_{{{\text{H}}3}}^{'}{{E}_{{\text{e}}}}; \\ \end{gathered} $$
(20)
$$\begin{gathered} {{\Omega }_{{{\mu \mu }}}} = {{\Omega }_{{{\tau L}}}} - \Omega _{{{\text{NL}}}}^{'};\,\,\Omega _{{{\text{NL}}}}^{'} = {{E_{{{\text{NL}}}}^{'}} \mathord{\left/ {\vphantom {{E_{{{\text{NL}}}}^{'}} {{{E}_{{{\text{H}}0}}}}}} \right. \kern-0em} {{{E}_{{{\text{H}}0}}}}}; \\ E_{{{\text{NL}}}}^{'} = N{\kern 1pt} '{{E}_{{\text{e}}}};\,\,(N{\kern 1pt} ' - N)\, \cdot {{{\chi }}_{0}} = n_{{\mu {\text{F}}}}^{'}; \\ \end{gathered} $$
(21)
$$\begin{gathered} {{\Omega }_{{{\mu \tau }}}} = {{\Omega }_{{{\tau L}}}} - 0.5({{\Omega }_{{{\text{HL}}}}} + {{\Omega }_{{{\text{GL}}}}}); \\ {{\Omega }_{{{\text{GL}}}}} = {{{{E}_{{{\text{GL}}}}}} \mathord{\left/ {\vphantom {{{{E}_{{{\text{GL}}}}}} {{{E}_{{{\text{H}}0}}}}}} \right. \kern-0em} {{{E}_{{{\text{H}}0}}}}};\,\,{{E}_{{{\text{GL}}}}} = {{n}_{{\text{G}}}}{{E}_{{\text{e}}}}. \\ \end{gathered} $$
(22)

Additional contributions \({{\Omega }_{{{\text{HL}}}}},\)\(\Omega _{{{\text{NL}}}}^{'},\) and \({{\Omega }_{{{\text{GL}}}}}\) are determined on the basis of energies \({{E}_{{{\text{HL}}}}},\)\(E_{{{\text{NL}}}}^{'},\)\({{E}_{{{\text{GL}}}}}\), and Higgs boson rest energy \({{E}_{{H0}}}.\) It follows from (20)–(22) that these additional energies are determined by quantum numbers \(n_{{{\text{H}}3}}^{'},\)\(N{\kern 1pt} ',\)\({{n}_{{\text{G}}}}\), and electron rest energy \({{E}_{{\text{e}}}}.\) At the same time,

$$\begin{gathered} n_{{{\text{H}}3}}^{'} = {{{{n}_{{{\text{H}}3}}}} \mathord{\left/ {\vphantom {{{{n}_{{{\text{H}}3}}}} {(1 + {{\Omega }_{{0{\nu }}}})}}} \right. \kern-0em} {(1 + {{\Omega }_{{0{\nu }}}})}}; \\ 1 + {{\Omega }_{{0{\nu }}}} = 1 + {{\left( {n_{{\text{F}}}^{'}} \right)}^{2}} = 1 + {{(N_{{\text{p}}}^{'} - N)}^{2}} \cdot {\chi }_{0}^{2}; \\ \end{gathered} $$
(23)
$$\begin{gathered} {{n}_{{{\text{H}}3}}} = {{Q}_{{{\text{H}}3}}}{{n}_{{{\text{h}}2}}} = 0.5{{Q}_{{{\text{H}}3}}}{{n}_{{A0}}}; \\ {{n}_{{A0}}} = {{z_{{\mu }}^{'}\left( {z_{{\mu }}^{'} + 1} \right) - {{n}_{{\text{Q}}}}} \mathord{\left/ {\vphantom {{z_{{\mu }}^{'}\left( {z_{{\mu }}^{'} + 1} \right) - {{n}_{{\text{Q}}}}} {{{n}_{{\text{g}}}}}}} \right. \kern-0em} {{{n}_{{\text{g}}}}}};\,\,{{n}_{{\text{Q}}}} = 2{{n}_{{\text{G}}}}. \\ \end{gathered} $$
(24)

Here, \({{{\chi }}_{0}} = 0.257104198\) is the effective susceptibility when the Higgs field is zero [12]; \({{n}_{{\text{g}}}} = 8,\)\({{n}_{{\text{Q}}}} = 6,\)\({{n}_{{\text{G}}}} = \left\langle {{{{\hat {c}}}_{{\text{G}}}}\,\hat {c}_{{\text{G}}}^{ + }} \right\rangle = 3\), and \(n_{{\text{G}}}^{'} = \left\langle {\hat {c}_{{\text{G}}}^{ + }\,{{{\hat {c}}}_{{\text{G}}}}} \right\rangle = 2\) can be interpreted as the numbers of the quanta of the gluon, quark, excited, and ground states of the gravitational field, respectively. Based on (24), we find \({{n}_{{{\text{H}}3}}} = 20.33926863.\) In light of (23), we obtain \(n_{{{\text{H}}3}}^{'} = 20.27965049,\) and \(N{\kern 1pt} ' = 17.21088699.\) Based on (20)–(22) we find energies \({{E}_{{{\text{HL}}}}}\) = \(10.36288254\,\,{\text{MeV}}{\text{,}}\)\(E_{{{\text{NL}}}}^{'}\) = \(8.794747246\,\,{\text{MeV}}{\text{,}}\) and \({{E}_{{{\text{GL}}}}} = \)\(1.53299721\,\,{\text{MeV;}}\) plus additional contributions \({{\Omega }_{{{\text{HL}}}}}\) = \(82.88159067 \times {{10}^{{ - 6}}},\)\(\Omega _{{{\text{NL}}}}^{'}\) = \(70.33975716 \times {{10}^{{ - 6}}},\) and \({{\Omega }_{{{\text{GL}}}}}\) = \(12.26080164 \times {{10}^{{ - 6}}}.\) Obtained parameters \(0.5{{\Omega }_{{{\mu e}}}}\) = \(1159.652705 \times {{10}^{{ - 6}}},\)\(0.5{{\Omega }_{{{\mu \mu }}}}\) = \(1165.923621 \times {{10}^{{ - 6}}},\) and \(0.5{{\Omega }_{{{\mu \tau }}}}\) = \(1177.307902 \times {{10}^{{ - 6}}}\) virtually coincide with the data in [15] for the anomalies of lepton magnetic moments.

Based on parameter \({{n}_{{{\text{H}}3}}},\) and spectral parameters \({{S}_{{iu}}}\) and \(S_{{0i}}^{'}\) from [12], we write the spectra for the occupational numbers: \({{n}_{{{\text{H}}ui}}} = 2{{n}_{{{\text{H}}3}}}{{S}_{{iu}}}\) and \({{n}_{{{\text{HS}}i}}} = 2{{n}_{{{\text{H}}3}}}S_{{0i}}^{'}.\) Spectrum branches \({{n}_{{{\text{H}}u4}}},\)\({{n}_{{{\text{H}}u3}}},\)\({{n}_{{{\text{HS4}}}}},\) and \({{n}_{{{\text{HS}}3}}}\) are optical, while \({{n}_{{{\text{H}}u2}}},\)\({{n}_{{{\text{H}}u1}}},\)\({{n}_{{{\text{HS}}2}}},\) and \({{n}_{{{\text{HS}}1}}}\) are acoustical. The laws of conservation are true for the spectrum branches:

$$\begin{gathered} {{n}_{{{\text{H}}3}}} = {{n}_{{{\text{H}}u4}}} - {{n}_{{{\text{H}}u2}}} = {{n}_{{{\text{H}}u3}}} + {{n}_{{{\text{H}}u1}}}; \\ {{n}_{{{\text{H}}3}}} = {{n}_{{{\text{HS}}4}}} - {{n}_{{{\text{HS}}2}}} = {{n}_{{{\text{HS}}3}}} + {{n}_{{{\text{HS}}1}}}. \\ \end{gathered} $$
(25)

The possibility of spectrum branch intersection follows from (25). Variations in lepton parameters \({{\Omega }_{{{\mu e}}}},\)\({{\Omega }_{{{\mu \mu }}}},\) and \({{\Omega }_{{{\mu \tau }}}};\) hyperfine splittings of atomic defects in fractal quantum systems; transitions from the discrete to the continuous energy spectrum; and manifestations of hystereses are possible.

CONCLUSIONS

Two models describing the supernonradiative states (SNSes) of different physical fields of active objects in fractal quantum systems were proposed on the basis of the theoretical Dicke superradiation model. It was shown that information on the presence of SNSes, dark matter, and dark relic photons can be extracted from the spectra of \({\gamma }\) radiation by detecting local minima against the background of stochastic signal behavior. Estimates of typical accelerations, temperatures, and energies were obtained for examples of active objects (nanoparticles, atomic defects, and neutrinos).

The possibility of describing mutual transformations of Bose- and Fermi-type particles was demonstrated using the description of neutrino oscillations. Using leptons as an example, magnetic moment anomalies were estimated with allowance for interconnections with the Higgs boson.

The results from this work can find application in neutrino physics, neuromedicine (to describe neuromediators), quantum information science, and quantum optical technologies.