Extended <Emphasis Type="Italic">E</Emphasis>(5) and <Emphasis Type="Italic">X</Emphasis>(5) symmetries: Series of models providing parameter-independent predictions

The E(5) symmetry describes nuclei related to the U(5)-SO(6) phase transition, while the X(5) symmetry is related to the U(5)-SU(3) phase transition. First, a chain of potentials interpolating between the U(5) symmetry of the five-dimensional harmonic oscillator and the E(5) symmetry is considered. Parameter-independent predictions for the spectra and B(E2) values of nuclei with R₄ = E(4)/E(2) ratios of 2.093, 2.135, and 2.157 (compared to the ratio of 2.000 of the U(5) case and the ratio of 2.199 of the E(5) case) are derived numerically and compared to existing experimental data, suggesting several new experiments. TheX(5) symmetry describes nuclei characterized byR₄=2.904.Using the same separation of variables of the original Bohr Hamiltonian as in X(5), an exactly soluble model with R₄=2.646 is constructed and its parameter-independent predictions are compared to existing spectra and B(E2) values. In addition, a chain of potentials interpolating between this new model and the X(5) symmetry is considered. Parameter-independent predictions for the spectra and B(E2) values of nuclei with R₄ ratios of 2.769, 2.824, and 2.852 are derived numerically and compared to existing experimental data, suggesting several new experiments.     

Ternary generalization of Pauli’s principle and the <Emphasis Type="Italic">Z</Emphasis><Subscript>6</Subscript>-graded algebras

We show how the discrete symmetries Z ₂ and Z ₃ combined with the superposition principle result in the SL(2,C) symmetry of quantum states. The role of Pauli’s exclusion principle in the derivation of the SL(2,C) symmetry is put forward as the source of the macroscopically observed Lorentz symmetry; then it is generalized for the case of the Z ₃ grading replacing the usual Z ₂ grading, leading to ternary commutation relations. We discuss the cubic and ternary generalizations of Grassmann algebra. Invariant cubic forms on such algebras are introduced, and it is shown how the SL(2,C) group arises naturally in the case of two generators only, as the symmetry group preserving these forms. The wave equation generalizing the Dirac operator to the Z ₃-graded case is introduced, whose diagonalization leads to a sixthorder equation. The solutions of this equation cannot propagate because their exponents always contain non-oscillating real damping factor. We show how certain cubic products can propagate nevertheless. The model suggests the origin of the color SU(3) symmetry.     

$$B\rightarrow K^*\mu ^+\mu ^-$$ optimised observables in the MSSM

We propose a unified extension of the standard model with three types of coexisting dark matter, motivated by three well-studied scenarios: (1) supersymmetry with \(R\) -parity conservation, (2) left–right symmetry where the \({SU}(2)_R\) doublet is not \((\nu ,e)_R\) but \((n,e)_R\) where \(n\) is a dark-matter candidate, and (3) a one-loop radiative neutrino mass model where the particles in the loop have an odd \(Z_2\) symmetry, the lightest of which is also a dark-matter candidate. Whereas there are many new particles in this model, the gauge couplings get unified in the same way as in the minimal supersymmetric standard model (MSSM). Since there are at least three dark-matter candidates, each one’s share of the total relic density of the Universe is adjustable, allowing for a much larger parameter space for the MSSM candidate itself. Our main focus is a careful study of a possible specific multipartite dark-matter scenario in this context.     

Implication of Spatial and Temporal Variations of the Fine-Structure Constant

Temporal and spatial variations of fine-structure constant \(\alpha \equiv e^{2}/\hbar c\) in cosmology have been reported in analysis of combination Keck and VLT data. This paper studies the variations based on consideration of basic spacetime symmetry in physics. Both laboratory α ₀ and distant α _z are deduced from relativistic spectrum equations of atoms (e.g., hydrogen atom) defined in inertial reference systems. When Einstein’s Λ≠0, the metric of local inertial reference systems in SM of cosmology is Beltrami metric instead of Minkowski, and the basic spacetime symmetry has to be de Sitter (dS) group. The corresponding special relativity (SR) is dS-SR. A model based on dS-SR is suggested. Comparing the predictions on α-varying with the data, the parameters are determined. The best-fit dipole mode in α’s spatial varying is reproduced by this dS-SR model. α-varyings in whole sky are also studied. The results are generally in agreement with the estimations of observations. The main conclusion is that the phenomenon of α-varying cosmologically with dipole mode dominating is due to the de Sitter (or anti de Sitter) spacetime symmetry with a Minkowski point in an extended special relativity called de Sitter invariant special relativity (dS-SR) developed by Dirac-Inönü-Wigner-Gürsey-Lee-Lu-Zou-Guo.     

Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents

We use the conformal bootstrap to perform a precision study of the operator spectrum of the critical 3d Ising model. We conjecture that the 3d Ising spectrum minimizes the central charge \(c\) in the space of unitary solutions to crossing symmetry. Because extremal solutions to crossing symmetry are uniquely determined, we are able to precisely reconstruct the first several \(\mathbb {Z}_2\) -even operator dimensions and their OPE coefficients. We observe that a sharp transition in the operator spectrum occurs at the 3d Ising dimension \(\Delta _\sigma = 0.518154(15)\) , and find strong numerical evidence that operators decouple from the spectrum as one approaches the 3d Ising point. We compare this behavior to the analogous situation in 2d, where the disappearance of operators can be understood in terms of degenerate Virasoro representations.     

Electroweak vacuum stability in classically conformal $$$$ extension of the standard model

We consider the minimal U(1)\(_{B-L}\) extension of the standard model (SM) with the classically conformal invariance, where an anomaly-free U(1)\(_{B-L}\) gauge symmetry is introduced along with three generations of right-handed neutrinos and a U(1)\(_{B-L}\) Higgs field. Because of the classically conformal symmetry, all dimensional parameters are forbidden. The \(B-L\) gauge symmetry is radiatively broken through the Coleman–Weinberg mechanism, generating the mass for the \(U(1)_{B-L}\) gauge boson (\(Z^\prime \) boson) and the right-handed neutrinos. Through a small negative coupling between the SM Higgs doublet and the \(B-L\) Higgs field, the negative mass term for the SM Higgs doublet is generated and the electroweak symmetry is broken. In this model context, we investigate the electroweak vacuum instability problem in the SM. It is well known that in the classically conformal U(1)\(_{B-L}\) extension of the SM, the electroweak vacuum remains unstable in the renormalization group analysis at the one-loop level. In this paper, we extend the analysis to the two-loop level, and perform parameter scans. We identify a parameter region which not only solve the vacuum instability problem, but also satisfy the recent ATLAS and CMS bounds from search for \(Z^\prime \) boson resonance at the LHC Run-2. Considering self-energy corrections to the SM Higgs doublet through the right-handed neutrinos and the \(Z^\prime \) boson, we derive the naturalness bound on the model parameters to realize the electroweak scale without fine-tunings.     

Orbital fractional parentage coefficients for the harmonic oscillator shell model

The Hamiltonian ofn particles moving in a common harmonic oscillator potential has as its symmetry group the unitary groupU(3n) in 3n dimensions,n particle states of the harmonic oscillator shell model can be characterized as bases of irreducible representations (BIR) of the groupU(3n) and of certain subgroups of this group. Use is made of these subgroups for the factorization and calculation of 2, 3, and 4 particle fractional parentage coefficients (fpc) of the harmonic oscillator shell model. Recoupling coefficients for subgroup chains of the symmetric groupS (n) appear as factors in the fpc. These coefficients are analyzed and calculated explicitly. The 2, 3 and 4 particle fpc of the 1s 1p shell configuration are obtained as products of these recoupling coefficients with known reduced Wigner coefficients of the unitary groupU(3) in 3 dimensions. Possible applications are indicated.     

Extended s-wave pairing symmetry on the triangular lattice heavy fermion system

We investigate the pairing symmetry of the Kondo-Heisenberg model on triangular lattice, which is believed to capture the core competition of Kondo screening and local magnetic exchange interaction in heavy electron compounds. On the dominant background of the heavy fermion state, the introduction of the Heisenberg antiferromagnetic interaction (J _H ) leads to superconducting pairing instability. Depending on the strength of the interactions, it is found that the pairing symmetry favours an extended s-wave for small J _H and high conduction electron density but a chiral \(d_{x^2 - y^2 } + id_{xy}\)-wave for large J _H and low conduction electron density, which provides a phase diagram of pairing symmetry from the calculations of the ground-state energy. The transition between these two pairing symmetries is found to be first-order. Furthermore, we also analyze the phase diagram from the pairing strengths and find that the phase diagram obtained is qualitatively consistent with that based on the ground-state energy. In addition, we propose an effective single-band BCS Hamiltonian, which is able to describe the low-energy thermodynamic behaviors of the heavy fermion superconducting states. These results further deepen the understanding of the antiferromagnetic interaction which results in a geometric frustration for the model studied. Our work may provide a possible scenario to understand the pairing symmetry of the heavy fermion superconductivity, which is one of active issues in very recent years.     

New constraints on the 3-3-1 model with right-handed neutrinos

In the framework of a 3-3-1 model with right-handed neutrinos and three scalar triplets we consider different spontaneous symmetry breaking patterns seeking for a non-linear realization of accidental symmetries of the model, which will produce physical Nambu–Goldstone (NG) bosons in the neutral scalar spectrum. We make a detailed study of the safety of the model concerning the NG boson emission in energy-loss processes which could affect the standard evolution of astrophysical objects. We consider the model with a \(\mathbb {Z}_2\) symmetry, conventionally used in the literature, finding that in all of the symmetry breaking patterns the model is excluded. Additionally, looking for solutions for that problem, we introduce soft \(\mathbb {Z}_2\)-breaking terms in the scalar potential in order to remove the extra accidental symmetries and at the same time maintain the model as simple as possible. We find that there is only one soft \(\mathbb {Z}_2\)-breaking term that enables us to get rid of the problematic NG bosons.     

Special features of the electric components of acoustic waves in the vicinity of nonpiezoactive directions in crystals

The conditions of existence of the zero components of electric field E and electric induction D accompanying a volume acoustic wave propagating in a piezoelectric medium have been studied. General equations describing the positions of the zero-field lines E(m) = 0 and the zero-induction points m₀, such that D(m₀) = 0 on the unit sphere (m² = 1) of the wave propagation directions, are obtained. General theorems determining the conditions ensuring the existence of such lines and points, even in triclinic crystals, are formulated. The relationship between such directions and various elements of the crystal symmetry is analyzed. The vector fields D(m), which are always orthogonal to the wave normals m, in the vicinity of the zero-induction points m₀ exhibit certain orientational singularities characterized by the Poincaré indices n = 0, ±1, ±2. The general analytical expressions are obtained for the n values in crystals with arbitrary anisotropy and specified for a number of crystals belonging to various symmetry classes. The conditions of stability of the orientational singularities with respect to small perturbations of the material moduli and a change in the crystal symmetry are considered.     

Scalar-leptoquark contributions to the neutrino magnetic moment

On the basis of astrophysical data on the neutrino magnetic moment, μ _ν < 3 × 10⁻¹² μ _B , constraints on the scalar-leptoquark masses are found within the minimal model involving four-color symmetry. It is shown that data on the neutrino magnetic moment are compatible with the mixing-parameter range that admits the existence of scalar leptoquarks whose masses are below 1 TeV, reaching experimental limits obtained from direct searches. In the case of mass degeneracy for the scalar leptoquarks S _m of electric charge Q = 2/3, the constraint m _{S m} > 330 GeV is obtained, which is independent of the mixing parameters of the model. The results are compared with the predictions of other leptoquark models. Original Russian Text ? A.V. Povarov, 2007, published in Yadernaya Fizika, 2007, Vol. 70, No. 5, pp. 905–911.     

Intensity of singlet-triplet transitions in C<Subscript>60</Subscript> fullerene calculated on the basis of the time-dependent density functional theory and taking into account the quadratic response

The singlet-singlet and triplet-triplet absorption spectra of C₆₀ fullerene are calculated using the density functional method and taking into account the theory of linear and quadratic responses. The B3LYP density functional and the 6–31G and 3–21G atomic basis sets are used. The calculations are performed using the D_2h and D_5d symmetry groups, although the real symmetry of the ground state is described by the I _h symmetry group. The matrix elements of the operator of the spin-orbit coupling are calculated and the probabilities of some singlet-triplet transitions are estimated. Taking into account the data in the literature on vibronic interactions of vibrations of the t_1u, t_2u, g _u , and h _u symmetry species, the radiative lifetime of the 1³T_2g → 1¹A _g phosphorescence was estimated to be 45 s. The fact that this time proved to be considerably greater than the experimentally observed total lifetime of the triplet testifies to a fast nonradiative deactivation of the lowest triplet state of C₆₀ fullerene and agrees with a low phosphorescence intensity. The zero-field splitting of some triplets and the intensities of magnetic dipole transitions are discussed.     

QED with vector and axial vector types of interaction and dynamical generation of particle masses

We consider a model of electrodynamics with two types of interaction, the vector \((e\bar \psi (\gamma ^\mu A_\mu )\psi )\) and axial vector \((e_A \bar \psi (\gamma ^\mu \gamma ^5 B_\mu )\psi )\) interactions, i.e., with two types of vector gauge fields, which corresponds to the local nature of the complete massless-fermion symmetry group U(1) ? U _A (1). We present a phenomenological model with spontaneous symmetry breaking through which the fermion and the axial vector field B_μ acquire masses. Based on an approximate solution of the Dyson equation for the fermion mass operator, we demonstrate the phenomenon of dynamical chiral symmetry breaking when the field B_μ has mass. We show the possibility of eliminating the axial anomalies in the model under consideration when introducing other types of fermions (quarks) within the standard-model fermion generations. We consider the polarization operator for the field B_μ and the procedure for removing divergences when calculating it. We demonstrate the emergence of a mass pole in the propagator of the particles that correspond to the field B_03BC when chiral symmetry is broken and consider the problems of regularizing closed fermion loops with axial vector vertices in connection with chiral symmetry breaking.     

Fermi Surface Topology in the Case of Spontaneously Broken Rotational Symmetry

The relation between the broken rotational symmetry of a system and the topology of its Fermi surface is studied for the two-dimensional system with the quasiparticle interaction f(p, p') having a sharp peak at |p ? p'| = q₀. It is shown that, in the case of attraction and q₀ = 2p_F the first instability manifesting itself with the growth of the interaction strength is the Pomeranchuk instability. This instability appearing in the L = 2 channel gives rise to a second order phase transition to a nematic phase. The Monte Carlo calculations demonstrate that this transition is followed by a sequence of the first and second order phase transitions corresponding to the changes in the symmetry and topology of the Fermi surface. In the case of repulsion and small values of q₀, the first transition is a topological transition to a state with the spontaneously broken rotational symmetry, namely, corresponding to the nucleation of L ? π(p_F/q₀ ? 1) small hole pockets at the distance p_F ? q₀ from the center and the deformation of the outer Fermi surface with the characteristic multipole number equal to L. At q₀ → 0, when the model under study transforms to the two-dimensional Nozières model, the multipole number characterizing the spontaneous deformation is L → ∞, whereas the infinitely folded Fermi curve acquires the Hausdorff dimension D = 2 which corresponds to the state with the fermion condensate.     

Is the CNMSSM more credible than the CMSSM?

“Post-sphaleron baryogenesis”, a fresh and profound mechanism of baryogenesis accounts for the matter–antimatter asymmetry of our present universe in a framework of Pati–Salam symmetry. We attempt here to embed this mechanism in a non-SUSY SO(10) grand unified theory by reviving a novel symmetry breaking chain with Pati–Salam symmetry as an intermediate symmetry breaking step and as well to address post-sphaleron baryogenesis and neutron–antineutron oscillation in a rational manner. The Pati–Salam symmetry based on the gauge group \(\mathrm{SU}(2)_L \times \mathrm{SU}(2)_{R} \times \mathrm{SU}(4)_C\) is realized in our model at \(10^{5}\) – \(10^{6}\)  GeV and the mixing time for the neutron–antineutron oscillation process having \(\Delta B=2\) is found to be \(\tau _{n-\bar{n}} \simeq 10^{8}\) – \(10^{10}\)  s with the model parameters, which is within the reach of forthcoming experiments. Other novel features of the model include low scale right-handed \(W^{\pm }_R\) , \(Z_R\) gauge bosons, explanation for neutrino oscillation data via the gauged inverse (or extended) seesaw mechanism and most importantly TeV scale color sextet scalar particles responsible for an observable \(n\) – \(\bar{n}\) oscillation which may be accessible to LHC. We also look after gauge coupling unification and an estimation of the proton lifetime with and without the addition of color sextet scalars.     

Superconducting pairing symmetry in frustrating t-J model: Electronic Raman absorption

Motivated by the controversy on the pairing symmetry of layered organic superconductors,we study electronic Raman scattering spectra on a frustrating lattice. A two-dimensionalt-t′-J-J′model and the Gutzwiller projectional variational method is used. The pairing symmetry isobtained self-consistently. Basing on this, we perform a systematic investigation of thedensity of states and electronic Raman spectra as a function oft′/t: ranging fromt′ = 0, the square lattice model, tot′ = t, the isotropic triangular latticemodel. We discuss the polarization dependence of the Raman spectra, which could be used toidentify the relevant superconducting pairing symmetry of frustrating systems such aslayered organic superconductors.     

Quantum groups as generalized gauge symmetries in WZNW models. Part II. The quantized model

This is the second part of a paper dealing with the “internal” (gauge) symmetry of the Wess–Zumino–Novikov–Witten (WZNW) model on a compact Lie group G. It contains a systematic exposition, for G = SU(n), of the canonical quantization based on the study of the classical model (performed in the first part) following the quantum group symmetric approach first advocated by L.D. Faddeev and collaborators. The internal symmetry of the quantized model is carried by the chiral WZNW zero modes satisfying quadratic exchange relations and an n-linear determinant condition. For generic values of the deformation parameter the Fock representation of the zero modes’ algebra gives rise to a model space of U _q (sl(n)). The relevant root of unity case is studied in detail for n = 2 when a “restricted” (finite dimensional) quotient quantum group is shown to appear in a natural way. The module structure of the zero modes’ Fock space provides a specific duality with the solutions of the Knizhnik–Zamolodchikov equation for the four point functions of primary fields suggesting the existence of an extended state space of logarithmic CFT type. Combining left and right zero modes (i.e., returning to the 2D model), the rational CFT structure shows up in a setting reminiscent to covariant quantization of gauge theories in which the restricted quantum group plays the role of a generalized gauge symmetry.     

In-plane torque and gap symmetry of untwinned YBa<Subscript>2</Subscript>Cu<Subscript>3</Subscript>O<Subscript>7</Subscript> crystals

The reversible magnetic torque of untwinned YBa₂Cu₃O₇ single crystals shows the four-fold symmetry in thea-b plane. The irreversible torque indicates evidence for a novel intrinsic pinning along thea andb axes. These facts mean that the free energy of the four-fold symmetry has a minimum when the field is applied along thea orb axis. The results are consistent with those expected from thed _x ²?y ² symmetry and rule out the possibility of thed _xy symmetry. The Fermi surface anisotropy is not responsible for the observed anisotropy. This is firstbulk evidence for thek-dependent gap anisotropy on the Fermi surface. The two-fold anisotropy parameter is found as\(\gamma _{ab} = \sqrt {{{m_a } \mathord{\left/ {\vphantom {{m_a } {m_b }}} \right. \kern-\nulldelimiterspace} {m_b }}} = 1.18 \pm 0.14\).     

Masses and Mixing of Neutral Leptons in a Grand Unified <Emphasis Type="Italic">E</Emphasis><Subscript>6</Subscript> Model with Intermediate Pati-Salam Symmetry

A brief review of the assignment of elementary fermions and bosons to irreducible multiplets in grand unified \(E_{6}\) models is followed by a discussion of different, hierarchical symmetry breaking chains from \(E_{6}\) down to \(SU(3)_{C} \times U(1)_{EM}\). We concentrate here on a model with an intermediate Pati-Salam symmetry for which \((B-L)\) is conserved. In particular, the mass/mixing matrix of electrically neutral fermions (i.e.neutrinos) that would be derived from Yukawa couplings is constructed. The pattern of neutrino masses and some bounds on mixing parameters are discussed.     

On hindered rotation of NO 2 − in KCl

Devonshire's model of a rigid dumb-bell subjected to a potential of octahedral symmetry is extended toC _2v rotors. As a practical example the low-lying energy states of NO ₂ ^? in KCl are evaluated and the allowed dipole transitions are compared with the experimental rotation-vibration spectrum at 15 °K.