Dark matter from the SU(4) model

The left-right symmetric Pati-Salam model of the unification of quarks and leptons is based on the SU(4) and SU(2)×SU(2) symmetry groups. These groups are naturally extended to include the classification of families of quarks and leptons. We assume that the family group (the group which unites the families) is also the SU(4) group. The properties of the fourth generation of fermions are the same as those of the ordinary-matter fermions in the first three generations except for the family charge of the SU(4)_F group: F=(1/3, 1/3, 1/3, ?1), where F=1/3 for fermions of ordinary matter and F=?1 for the fourth-generation fermions. The difference in F does not allow mixing between ordinary and fourth-generation fermions. Because of the conservation of the Fcharge, the creation of baryons and leptons in the process of electroweak baryogenesis must be accompanied by the creation of fermions of the fourth generation. As a result, the excess n _B of baryons over antibaryons leads to the excess n_4ν=N?N? of neutrinos over antineutrinos in the fourth generation with n_4ν=n _B . This massive neutrino may form nonbaryonic dark matter. In principle, the mass density of the fourth neutrino n_4νm _N in the Universe can make the main contribution to dark matter, since the lower bound on the neutrino mass m _N from the data on decay of the Z bosons is m _N <m _Z /2. The straightforward prediction of this model leads to the amount of cold dark matter relative to baryons, which is an order of magnitude higher than allowed by observations. This inconsistency may be avoided by nonconservation of the F charge.     

New neutron magic number <Emphasis Type="Italic">N</Emphasis> = 16 for neutron-rich nuclei

A survey of experimental results obtained at GANIL (Caen, France) on the study of the properties of very neutron-rich nuclei (Z=6–20, A=20–60) near the neutron drip line and resulting in an appearance of further evidence for the new magic number N=16 is presented. Very recent data on mass measurements of neutron-rich nuclei at GANIL and some characteristics of binding energies in this region are discussed. Nuclear binding energies are very sensitive to the existence of nuclear shells, and together with the measurements of instability of doubly magic nuclei ¹⁰He and ²⁸O, they provide information on changes in neutron shell closures of very neutron-rich isotopes. The behavior of the two-neutron separation energies S_2n derived from mass measurements gives very clear evidence for the existence of the new shell closure N=16 for Z=9 and 10 appearing between the 2s_1/2 and 1d_3/2 orbitals. This fact, strongly supported by the instability of C, N, and O isotopes with N>16, confirms the magic character of N=16 for the region from carbon up to neon, while the shell closure at N=20 tends to disappear for Z≤13. Decay studies of these hardly accessible short-lived neutron-rich nuclei from oxygen to silicon using in-beam γ-ray spectroscopy are also reported.     

Electron gas induced in SrTiO<Subscript>3</Subscript>

This mini-review is dedicated to the 85th birthday of Prof. L.V. Keldysh, from whom we have learned so much. In this paper, we study the potential and electron density depth profiles in surface accumulation layers in crystals with a large and nonlinear dielectric response such as SrTiO₃ (STO) in the cases of planar, spherical, and cylindrical geometries. The electron gas can be created by applying an induction D₀ to the STO surface. We describe the lattice dielectric response of STO using the Landau–Ginzburg free energy expansion and employ the Thomas–Fermi (TF) approximation for the electron gas. For the planar geometry, we arrive at the electron density profile n(x) ∝ (x + d)^–12/7, where d ∝ D₀^–12/7. We extend our results to overlapping electron gases in GTO/STO/GTO heterojunctions and electron gases created by spill-out from NSTO (heavily n-type doped STO) layers into STO. Generalization of our approach to a spherical donor cluster creating a big TF atom with electrons in STO brings us to the problem of supercharged nuclei. It is known that for an atom with a nuclear charge Ze where Z > 170, electrons collapse onto the nucleus, resulting in a net charge Zn < Z. Here, instead of relativistic physics, the collapse is caused by the nonlinear dielectric response. Electrons collapse into the charged spherical donor cluster with radius R when its total charge number Z exceeds the critical value Z_c ≈ R/a, where a is the lattice constant. The net charge eZ_n grows with Z until Z exceeds Z* ≈ (R/a)^9/7. After this point, the charge number of the compact core Z_n remains ≈ Z*, with the rest Z* electrons forming a sparse TF atom with it. We extend our studies of collapse to the case of long cylindrical clusters as well.     

<Emphasis Type="Italic">Z</Emphasis><Superscript>0</Superscript>-boson decays in a strong electromagnetic field

The probability of Z ⁰-boson decay to a pair of charged fermions in a strong electromagnetic field, Z ⁰ → \(\bar f\) f, is calculated. On the basis of a method that employs exact solutions to relativistic wave equations for charged particles, an analytic expression for the partial decay width Γ(?) = Γ(Z ⁰ → \(\bar f\) f) is obtained at an arbitrary value of the parameter ? = \(eM_Z^{ - 3} \sqrt { - (F_{\mu \nu } q^\nu )^2 } \), which characterizes the external-field strength. The total Z ⁰-boson decay width in an intense electromagnetic field, Γ _Z (?), is calculated by summing these results over all known generations of charged leptons and quarks. It is found that, in the region of relatively weak fields (? < 0.06), the field-induced corrections to the standard Z ⁰-boson decay width in a vacuum do not exceed 2%. As ? increases, the total decay width Γ _Z (?) develops oscillations against the background of its gradual decrease to the absolute-minimum point. At ?_min = 0.445, the total Z ⁰-boson decay width reaches the minimum value of Γ _Z (?_min) = 2.164 GeV, which is smaller than the Z ⁰-boson decay width in a vacuum by more than 10%. In the region of superstrong fields (? > 1), Γ _Z (?) grows monotonically with increasing external-field strength. In the region ? > 5, the t-quark-production process Z ⁰ → \(\bar t\) t, which is forbidden in the absence of an external field, begins contributing significantly to the total decay width of the Z ⁰ boson.     

New Bounds on the Maximum Ionization of Atoms

We prove that the maximum number N _c of non-relativistic electrons that a nucleus of charge Z can bind is less than 1.22Z + 3Z ^1/3. This improves Lieb’s upper bound N _c  < 2Z + 1 Lieb (Phys Rev A 29:3018–3028, 1984) when Z ≥ 6. Our method also applies to non-relativistic atoms in magnetic field and to pseudo-relativistic atoms. We show that in these cases, under appropriate conditions, \({\limsup_{Z \to \infty}N_c/Z \le 1.22}\).     

Coulomb Problem for <Emphasis Type="Italic">Z</Emphasis> &gt; Z<Subscript>cr</Subscript> in Doped Graphene

The dynamics of charge carriers in doped graphene, i.e., graphene with a gap in the energy spectrum depending on the substrate, in the presence of a Coulomb impurity with charge Z is considered within the effective two-dimensional Dirac equation. The wave functions of carriers with conserved angular momentum J = M + 1/2 are determined for a Coulomb potential modified at small distances. This case, just as any two-dimensional physical system, admits both integer and half-integer quantization of the orbital angular momentum in plane, M = 0, ±1, ±2, …. For J = 0, ±1/2, ±1, critical values of the effective charge Z_cr(J, n) are calculated for which a level with angular momentum J and radial quantum numbers n = 0 and n = 1 reaches the upper boundary of the valence band. For Z < Z_cr (J, n = 0), the energy of a level is presented as a function of charge Z for the lowest values of orbital angular momentum M, the level with J = 0 being the first to descend to the band edge. For Z>Z_cr (J, n = 0), scattering phases are calculated as a function of hole energy for several values of supercriticality, as well as the positions ε₀ and widths γ of quasistationary states as a function of supercriticality. The values of ε₀* and width γ* are pointed out for which quasidiscrete levels may show up as Breit–Wigner resonances in the scattering of holes by a supercritical impurity. Since the phases are real, the partial scattering matrix is unitary, so that the radial Dirac equation is consistent even for Z > Z_cr. In this single-particle approximation, there is no spontaneous creation of electron–hole pairs, and the impurity charge cannot be screened by this mechanism.     

Dynamical screening of AMM and QED effects for large-<Emphasis Type="Italic">Z</Emphasis> hydrogen-like atoms

The effective interaction ΔU_AMM of the anomalous magnetic moment (AMM) of an electron with the Coulomb field of an extended nucleus is analyzed. As soon as the q² dependence of the electron formfactor F₂(q²)is taken into account from the beginning, the AMM is found to be dynamically screened at small distances of r ? 1/m. The ΔU_AMM effects on the low-lying electronic levels of a superheavy extended nucleus with Zα > 1are analyzed using the nonperturbative approach. The growth rate of the ΔU_AMM contribution with increasing Z is shown to be essentially nonmonotonic. At the same time, the energy shifts of electronic levels in the vicinity of the threshold of the lower continuum monotonically decrease in the region Z ?Z_cr,1s. The latter result is generalized to the whole self-energy contribution to energy shifts of electronic levels, thus also referring to the possible behavior of QED radiative effects with virtual-photon exchange, considered beyond the framework of the perturbative expansion in Zα.     

Single-particle properties of <Emphasis Type="Italic">N</Emphasis> = 12 to <Emphasis Type="Italic">N</Emphasis> = 20 silicon isotopes within the dispersive optical model

Experimental neutron and proton single-particle energies in N = 12 to N = 20 silicon isotopes and data on neutron and proton scattering by nuclei of the isotope ²⁸Si are analyzed on the basis of the dispersive optical model. Good agreement with available experimental data was attained. The occupation probabilities calculated for the single-particle states in question suggest a parallel-type filling of the 1d and 2s _1/2 neutron states in the isotopes ^{26,28,30,32,34}Si. The single-particle spectra being considered are indicative of the closure of the Z = 14 proton subshell in the isotopes ^30,32,34Si and the N = 20 neutron shell.     

Optical and x-ray diffraction studies of the symmetry of distorted phases of the (NH<Subscript>4</Subscript>)<Subscript>3</Subscript>ZrF<Subscript>7</Subscript> crystal

(NH₄)₃ZrF₇ single crystals were grown, and polarization-optical and x-ray diffraction studies were performed on powders and crystalline plates of various cuts over a wide temperature range. Phase transitions are revealed at temperatures T _1↑ = 280 K, T _2↑ = 279.6 K, T _3↑ = 260–265 K, and T _4↑ = 238 K on heating and at T _1↓ = 280 K, T _2↓ = 269–270 K, T _3↓ = 246 K, and T _4↓ = 235 K on cooling. The sequence of changes in symmetry is established to be as follows: O _h ⁵ (Z = 4) ? D _2h ²⁵ (Z = 2) ? C _2h ³ (Z = 2) ? C _i ¹ (Z = 108) ? monoclinic²(Z = 216).     

Multicomponent <Emphasis Type="Italic">n</Emphasis>-(Bi,Sb)<Subscript>2</Subscript>(Te,Se,S)<Subscript>3</Subscript> solid solutions with different atomic substitutions in the Bi and Te sublattices

The thermoelectric properties of n-Bi_{2 ? x} Sb _x Te_{3 ? y ? z} Se _y S _z solid solutions are studied in the temperature range 300–550 K. It is shown that an increase in the parameter β determining the figure-of-merit Z of the material is observed in compositions with the optimally related effective mass of the density of states m/m ₀, the carrier mobility μ₀, and the lattice thermal conductivity κ _L . Within the temperature range 300–350 K, the parameter β and the figure-of-merit Z are found to increase in solid solutions with substitutions in both bismuth telluride sublattices Bi → Sb and Te → Se, S (x = 0.16, y = z = 0.12) for optimum electron concentrations. An increase in the electron concentration and substitutions of atoms only in the tellurium sublattice bring about an increase in the β parameter and the value of Z at higher temperatures. Within the range 350–450 K, the parameters β and Z are observed to increase in a solid solution with a low content of substituted atoms in the tellurium sublattice Te → Se, S for y = z = 0.09 and, at higher temperatures up to 550 K, in compositions with tellurium substituted by selenium only, with increasing content of substituted atoms.     

Screened self-energy correction to the 2<Emphasis Type="Italic">p</Emphasis><Subscript>3/2</Subscript>-2<Emphasis Type="Italic">s</Emphasis> transition energy in Li-like ions

We present an ab initio calculation of the screened self-energy correction for 1s² 2p_3/2 and 1s² 2s states of Li-like ions with nuclear charge numbers in the range Z = 12?100. The evaluation is carried out to all orders in the nuclear strength parameter Zα. This investigation concludes our calculations of all two-electron QED corrections for the 2p_3/2-2s transition energy in Li-like ions and thus considerably improves theoretical predictions for this transition for high-Z ions.     

Running electromagnetic-coupling constant: Low-energy normalization and the value at <Emphasis Type="Italic">M</Emphasis><Subscript><Emphasis Type="Italic">Z</Emphasis></Subscript>

A numerical value for the running electromagnetic-coupling constant in the \(\overline {MS} \) scheme is calculated at a low-energy normalization scale equal to the τ-lepton mass M_τ. This low-energy boundary value is used for running the electromagnetic coupling to larger scales, where high-precision experimental measurements can be performed. Particular scales of interest are the b-quark mass for studying ?-resonance physics and the Z-boson mass M _Z for high-precision tests of the Standard Model and for the determination of the Higgs boson mass from radiative corrections. A numerical value of the running electromagnetic-coupling constant at M _Z in the on-shell renormalization scheme is also given.     

Special features of the structure of the beta-stability line for nuclei

The mass-number (A) dependence Z _β(A) for nuclei lying on the beta-stability line (BSL) is calculated for A and Z values in the ranges of A = 2–258 and Z = 1–100, respectively. The calculated values are compared with experimental data. The deviations ΔZ = Z _expt ? Z _β are analyzed. This analysis of ΔZ reveals that there are three regions of A values in which the A dependence of ΔZ is parabolic. The possible forms of the A dependence of ΔZ are analyzed, and it is shown that the majority of nuclei belong to several parabolas simultaneously.     

Spectra and mean energies of prompt neutrons from <Superscript>238</Superscript>U fission induced by primary neutrons of energy in the region <Emphasis Type="Italic">E</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript><20 MeV

The time-of-flight technique is used to measure the ratios R(E, E _n )=N(E, E _n )/N_Cf(E) of the normalized (to unity) spectra N(E, E _n ) of neutrons accompanying the neutron-induced fission of ²³⁸U at primary-neutron energies of E _n =6.0 and 7.0 MeV to the spectrum N_Cf(E) neutrons from the spontaneous fission of ²⁵²Cf. These experimental data and the results of their analysis are discussed together with data that were previously obtained for the neutron-induced fission of ²³⁸U at the primary energies of E _n =2.9, 5.0, 13.2, 14.7, 16.0, and 17.7 MeV.     

New magic nuclei and conditions of their formation

The systematic studies of the arrangement features of single-particle nucleon subshells in even-even ^90,92,94,96Zr isotopes and behavior of some known “magicity parameters” in isotopes and isotones neighboring the ⁹⁶Zr nucleus have led to the interpretation of ⁹⁶Zr as a new doubly magic nucleus. Analysis of the structure of nucleon shells in the ⁹⁶Zr nucleus revealed a feature, which consisted in that near the Fermi energy it had filled proton (π1f _5/2) and neutron (v2d _5/2) subshells with an identical and large total momentum j = 5/2, which was called the j-j coupling. Above the π1f _5/2 shell, there is another filled shell (π2p _1/2) with two j = 1/2 protons. Applied to other filled shells, this empirical rule allowed revealing several new nontraditional magic nuclei: ⁹⁶Sr (Z = 38, N = 58), π1f _5/2, v2d _5/2, and v3s _1/2 subshells; ⁵⁴Ca (Z = 20, N = 34), π2p _3/2, v1d _3/2, and v2p _1/2 subshells; a pair of ³⁰Si (Z = 14, N = 16) and ³⁰S (Z = 16, N = 14) nuclei, π1d _5/2, v1d _5/2, and (π/v)2s _1/2 subshells; and a pair of ¹⁴C (Z = 6, N = 8) and ¹⁴O (Z = 8, N = 6) nuclei, 1p _3/2, v1p _3/2, and v2p _1/2 subshells. The existence of the magic nuclei ^52,54Ca is widely discussed in the literature, the possibility of the existence of the other nuclei found is confirmed by the systematics of the behavior of the “magicity” parameters. The fact that shells with some nucleon numbers different from the classical magic numbers are closed may be due to the manifestation of a new type of interaction between nuclear protons and neutrons occupying certain subshells.     

Probing of <Emphasis Type="Italic">Z</Emphasis>γγ production sensitivity to anomalous quartic gauge couplings at LHC experiments with √ <Emphasis Type="Italic">s</Emphasis> =13 TeV

Sensitivity to anomalous ZZγγ and Zγγγ couplings in Zγγ production was probed for the ATLAS experiment at Large Hadron Collider. Zγγ process with anomalous couplings simulation in ppcollisions with √ s = 13 TeV was performed using VBFNLO MC generator. The expected limits on the Effective Field Theory parameters f _T0/Λ4, f _T5/Λ4, f _T9/Λ4, f _M2/Λ4, f _M3/Λ4 were extracted for 5 fb^?1 integral luminosity using the distribution on the invariant mass of Zγγ from the combination of charged leptonic decay channels of Z boson (Zγγ → μ⁺μ^?γγ and Zγγ → e⁺e^?γγ).     

Photon distribution functions of the fluorescence photons from a single nanoparticle in various photon counting methods

Simple expressions have been derived for three photon distribution functions w _N ^M (T), w _N ^Z (T), and w _N ^O (T) corresponding to three different methods for counting fluorescence photons from a single nanoparticle excited by continuous laser radiation. In contrast to the previously derived expressions represented in the form of N multiple integrals, the new expressions contain only single or double integrals of Poisson functions, which makes it possible to easily perform the numerical calculation of the photon distribution. The simplest photon counting method corresponds to the lengthiest function w _N ^M (T); on the contrary, the simplest function w _N ^O (T) corresponds to the most complex photon counting method. The functions w _N ^M (T), w _N ^Z (T), and w _N ^O (T) are noticeably different in short time intervals T; however, the distributions calculated using these functions are almost indistinguishable from each other in long T intervals. This circumstance makes it possible to use the simplest function w _N ^O (T) to consider the photon statistics measured by the simplest method. This possibility is particularly important for investigating the fluorescence photon statistics, where the intensity fluctuates.     

Chiral and deconfinement phase transitions in QED<Subscript>3</Subscript> with finite gauge boson mass

Based on the experimental observation that there is a coexisting region between the antiferromagnetic (AF) and d-wave superconducting (dSC) phases, the influences of gauge boson mass m _a on chiral symmetry restoration and deconfinement phase transitions in QED₃ are investigated simultaneously within a unified framework, i.e., Dyson–Schwinger equations. The results show that the chiral symmetry restoration phase transition in the presence of the gauge boson mass m _a is a typical second-order phase transition; the chiral symmetry restoration and deconfinement phase transitions are coincident; the critical number of fermion flavors N ^c _f decreases as the gauge boson mass m _a increases, which implies that there exists a boundary that separates the N ^c _f –m _a plane into chiral symmetry breaking/confinement region for (N ^c _f , m _a ) below the boundary and chiral symmetry restoration/deconfinement region for (N ^c _f , m _a ) above it.     

Calculation of<Emphasis Type="Italic">Z</Emphasis><Subscript>eff</Subscript> from plasma resistivity in IR-T1 tokamak

The effective ion charge,Z _eff, represents the average charge of ionsZ _i of gasses inside the system, which indicates the level of the impurities in the plasma. Several techniques have been applied to estimateZ _eff, like mass spectroscopy, anomaly factor and Bremsstrahlung radiation.

We estimatedZ _eff in the IR-T1 tokamak through anomaly factor. The IR-T1 tokamak is a small air-core transformer tokamak with circular cross section and with out conducting shell and divertor. Its aspect ratio is

$$\frac{R}{a} = \frac{{45 cm}}{{12.5 cm}}.$$

For a tokamak discharge of 30 kA plasma current and 1.5 V of loop voltage and by anomaly factor we observed thatZ _eff value is about 1.5.     

Collective states in highly symmetric atomic configurations and single-photon traps

We study correlated states in circular and linear-chain configurations of identical two-level atoms containing the energy of a single quasi-resonant photon in the form of a collective excitation, where the collective behavior is mediated by exchange of transverse photons between the atoms. For a circular atomic configuration containing N atoms, the collective energy eigenstates can be determined by group-theoretical means making use of the fact that the configuration possesses a cyclic symmetry group Z _N . For these circular configurations, the carrier spaces of the various irreducible representations of the symmetry group are at most two-dimensional, so that the effective Hamiltonian on the radiationless subspace of the system can be diagonalized analytically. As a consequence, the radiationless energy eigenstates carry a Z _N quantum number p = 0, 1, …, N, which is analogous to the angular momentum quantum number l = 0, 1, … carried by particles propagating in a central potential, such as a hydrogen-like system. Just as the hydrogen s states are the only electronic wave functions that can occupy the central region of the Coulomb potential, the quasi-particle corresponding to a collective excitation of the circular atomic sample can occupy the central atom only for vanishing Z _N quantum number p. When a central atom is present, the p = 0 state splits into two, showing level crossing at certain radii; in the regions between these radii, damped oscillations between two “ extreme” p = 0 states occur, where the excitation occupies either the outer atoms or the central atom only. For large numbers of atoms in a maximally subradiant state, a critical interatomic distance of λ/2 emerges both in the linear-chain and in the circular configuration of atoms. The spontaneous decay rate of the linear configuration exhibits a jumplike “critical” behavior for next-neighbor distances close to a half-wavelength. Furthermore, both the linear-chain and the circular configurations exhibit exponential photon trapping once the next-neighbor distance becomes less than a half-wavelength, with the suppression of spontaneous decay being particularly pronounced in the circular system. In this way, circular configurations containing sufficiently many atoms may be natural candidates for single-photon traps.