About Group Representations in Grand Unified Theories

The main purpose of this article is to systematize group theory methods in Grand Unified Theories (GUTs). An exact representation structure for all admissible groups that unify elementary forces is analysed. The implications of requirements asymptotic freedom for standard SU (3)_c ⊗ SU(2)_L ⊗ U(1)_y model is investigated in detail. It is shown how restrictive are such assumptions in unified model building. As a result the maximal number of families is five independently on the kind of considered model. In several cases this number is even smaller. The maximal horizontal structure of interactions which is possible to embed into unification scheme is also discussed for all admissible models. The number of superheavy particles is evaluated in different unifying schemes.     

Quaternion Octonion Reformulation of Grand Unified Theories

In this paper, Grand Unified theories are discussed in terms of quaternions and octonions by using the relation between quaternion basis elements with Pauli matrices and Octonions with Gell Mann λ matrices. Connection between the unitary groups of GUTs and the normed division algebra has been established to re-describe the SU(5) gauge group. We have thus described the SU(5) gauge group and its subgroup SU(3) _C ×SU(2) _L ×U(1) by using quaternion and octonion basis elements. As such the connection between U(1) gauge group and complex number, SU(2) gauge group and quaternions and SU(3) and octonions is established. It is concluded that the division algebra approach to the theory of unification of fundamental interactions as the case of GUTs leads to the consequences towards the new understanding of these theories which incorporate the existence of magnetic monopole and dyon.     

Identifying an extraZ 0 boson with a new model

According to the principle of minimality, we find a newSU(6) model. ThisSU(6) model, and other models, can be identified as a theoretical origin of an extra Z⁰ boson. We apply the strategy of Boudjemaet al. (BLRV) which is very effective in identifying the theoretical origin of an extra Z⁰ boson in the newSU(6) model, and compare the model with six other models.     

The Mass of the Ninth Pseudoscalar Meson in the Frame of Broken Chiral SU(3) ⊗ SU(3)

Using GELL -MANN 's ansatz for the SU(3)?SU(3) symmetry breaking part H_SB = -u⁰ -cu⁸ in the strong HAMILTONIAN density, where the operators u^j (j = 0, 1,…8) are the scalar part of a basis for the {(3,3) ⊕ (3,3)} representation of chiral SU(3)?SU(3) and where the constant c is a measure for SU(3) breaking within the SU(3)?SU(3) breaking, a sum rule for the spin zero spectral functions of the pseudoscalar axial vector current octet is derived. Saturating the sum rule with the lowest lying states, the mass of the ninth pseudoscalar meson can be estimated as m_η1 = 950 MeV.     

Spectral Measures and Generating Series for Nimrep Graphs in Subfactor Theory II: <Emphasis Type="Italic">SU</Emphasis>(3)

We complete the computation of spectral measures for SU(3) nimrep graphs arising in subfactor theory, namely the SU(3) ADE{SU(3) \mathcal{ADE}} graphs associated with SU(3) modular invariants and the McKay graphs of finite subgroups of SU(3). For the SU(2) graphs the spectral measures distill onto very special subsets of the semicircle/circle, whilst for the SU(3) graphs the spectral measures distill onto very special subsets of the discoid/torus. The theory of nimreps allows us to compute these measures precisely. We have previously determined spectral measures for some nimrep graphs arising in subfactor theory, particularly those associated with all SU(2) modular invariants, all subgroups of SU(2), the torus \mathbbT², SU(3){\mathbb{T}^2,\,SU(3)}, and some SU(3) graphs.     

A spectrum generating algebra for meson resonances

The algebraSO(6,1) is considered as a unification ofSO(6), which is isomorphic toSU(4) SU(3), and the de Sitter algebraSO(4,1). The latter replaces the Poincaré algebra as the algebra of the group of motions of physical space-time. A representation ofSO(6,1) is constructed, which, on restriction toSU(3), decomposes into the direct sum of allSU(3) representations, each occurring just once in the decomposition. The expectation values of the mass-squared operator, when evaluated in the octet, give accurate mass formulae for the octets of 1^– and 2⁺ meson resonances.     

Quaternionic-valued Gravitation in 8D, Grand Unification and Finsler Geometry

A unification model of 4D gravity and SU(3)×SU(2)×U(1) Yang-Mills theory is presented. It is obtained from a Kaluza-Klein compactification of 8D quaternionic gravity on an internal CP ²=SU(3)/U(2) symmetric space. We proceed to explore the nonlinear connection formalism used in Finsler geometry to show how ordinary gravity in D=4+2 dimensions has enough degrees of freedom to encode a 4D gravitational and SU(5) Yang-Mills theory. This occurs when the internal two-dim space is a sphere S ². This is an appealing result because SU(5) is one of the candidate GUT groups. We conclude by discussing how the nonlinear connection formalism of Finsler geometry provides an infinite hierarchical extension of the Standard Model within a six dimensional gravitational theory due to the embedding of SU(3)×SU(2)×U(1)⊂SU(5)⊂SU(∞).     

SU(2)×SU(2) Electroweak Theory I: The B3 Field on the Physical Vacuum

Nonlinear optics confronts the U(1) theory of electrodynamics with the dilemma of the existence of nonlinear fields. The U(1) group is completely linear and Abelian and causes consideration of an SU(2) theory of electrodynamics. An SU(2) theory of electrodynamics, with a B ³ magnetic field, means that physics is forced to consider an SU(2) × SU(2) electroweak theory. It is then demonstrated that the B ³ field exists on the physical vacuum defined by the Higgs symmetry breaking of this extended electroweak theory.     

Flavored surface defects in 4d $$\mathcal{N}=1$$ N = 1 SCFTs

We discuss supersymmetric surface defects in compactifications of six-dimensional minimal conformal matter of types SU(3) and SO(8) to four dimensions. The relevant field theories in four dimensions are \(\mathcal{N}=1\) quiver gauge theories with SU(3) and SU(4) gauge groups, respectively. The defects are engineered by giving space-time-dependent vacuum expectation values to baryonic operators. We find evidence that in the case of SU(3) minimal conformal matter, the defects carry SU(2) flavor symmetry which is not a symmetry of the four-dimensional model. The simplest case of a model in this class is SU(3) SQCD with nine flavors, and thus the results suggest that this admits natural surface defects with SU(2) flavor symmetry. We analyze the defects using the superconformal index and derive analytic difference operators introducing the defects into the index computation. The duality properties of the four-dimensional theories imply that the index of the models is a kernel function for such difference operators. In turn, checking the kernel property constitutes an independent check of the dualities and the dictionary between six- dimensional compactifications and four-dimensional models.

     

Hierarchy of symmetry breakings and neutral currents in anSU(6) grand unification model

In anSU(6) grand unification model with eight quarks and eight leptons belonging to 15-plet and singlet representations, the symmetry is spontaneously broken by the sequenceSU(6)SU(3) ^c ×SU(2)×U(1)×U(1)SU(3) ^c ×U(1). Fror two cases of symmetry breakings the effective weak neutral current coupling constants are compared with experiment. For theSU(3) ^c ×SU(2)×U(1)×U(1)×SU(3) ^c ×U(1) symmetry breaking, the coupling constants reproduce the Weinberg-Salam model with a small correction term. Agreement with the experimental mean values is improved with the correction term. Parity violation in atomic physics is also discussed.     

A note about the t’Hooft ansatz for SU(N) real-time gauge theories

The t’Hooft ansatz, which reduces the classical Yang-Mills theory to the λϕ⁴ one, is under consideration. It is shown that, in the framework of this ansatz, the real-time classical solutions for the arbitrary SU(N) gauge group are obtained by embedding SU(2) × SU(2) into SU(N). It is argued that this group structure is the only possibility in the framework of the considered ansatz. New explicit solutions for SU(3) and SU(5) gauge groups are shown. The text was submitted by the authors in English.     

New physics contribution to neutral trilinear gauge boson couplings

We study the one-loop new physics effects to the CP even triple neutral gauge boson vertices γ ^⋆ γ Z, γ ^⋆ Z Z, Z ^⋆ Z γ and Z ^⋆ ZZ in the context of Little Higgs models. We compute the contribution of the additional fermions in Little Higgs models in the framework of direct product groups where [SU(2)×U(1)]² gauge symmetry is embedded in SU(5) global symmetry and also in the framework of the simple group where SU(N)×U(1) gauge symmetry breaks down to SU(2) _L ×U(1). We calculate the contribution of the fermions to these couplings when T parity is invoked. In addition, we re-examine the MSSM contribution at the chosen point of SPS1a′ and compare with the SM and Little Higgs models.     

Dynamical and symmetry groups of the SU(2) Kepler problem

It is well known that the MIC–Kepler problem, an extension of the three-dimensional Kepler problems, admits the same dynamical and symmetry groups as the Kepler problem. This paper aims to study dynamical and symmetry groups of the SU(2) Kepler problem, where the SU(2) Kepler problem is defined to be the dynamical system reduced from the eight-dimensional conformal Kepler problem through an SU(2) symmetry and turns out to be an extension of the five-dimensional Kepler problem. It is shown that the SU(2) Kepler problem admits a dynamical group SO^*(8) and that the phase space of the SU(2) Kepler problem is symplectomorphic with a co-adjoint orbit of SO^*(8), on which the Kirillov–Kostant–Souriau form is defined. It is further shown that the subgroups, SU(4), SU^*(4), and Sp(2)×_SR⁵, of SO^*(8) provide the symmetry groups, SU(4)/Z₂SO(6), SU^*(4)/Z₂SO₀(1,5), and (Sp(2)×_SR⁵)/Z₂SO(5)×_SR⁵, of the SU(2) Kepler problem with negative, positive, and zero energies, respectively, where ×_S denotes a semi-direct product. Furthermore, constants of motion for the SU(2) Kepler problem are found together with their Poisson brackets. The symmetry Lie algebra formed by constants of motion is shown to be isomorphic with so(6)su(4), so(1,5)su^*(4), or so(5)_SR⁵sp(2)_SR⁵, depending on whether the energy is negative, positive, or zero, where _S denotes a semi-direct sum. These Lie algebras are subalgebras of so^*(8)so(2,6).     

Toward a universal Lagrangian

A gauge theory with the gauge groupU(1)×U(1)×SU(2)×SU(3)×SU(4) is shown to fit well into the generalized Kaluza scheme with eleven-dimensional space-time and its compact subspaceS ²×S ⁵. A unified theory is obtained which exhibits some broken super-symmetric features (N = 8). Our approach is dictated by phenomenological requirements. The appearance of three generations of leptons and six flavors of colored quarks follows naturally. Within our Lagrangian there appear several free parameters (coupling constants), but some relations between them may follow from the requirement of cancellation of divergencies.Temporarily at Dublin Institute for Advanced Studies, School of Theoretical Physics, 10 Burlington Rd., Dublin 4, Ireland.     

Finite unified theories and their predicitions

All-loop Finite Unified Theories (FUTs) are very interesting N = 1 supersymmetric Grand Unified Theories (GUTs) realising an old field theory dream, and moreover have a remarkable predictive power due to the required reduction of couplings. The reduction of the dimensionless couplings in N = 1 GUTs is achieved by searching for renormalization group invariant (RGI) relations among them holding beyond the unification scale. Finiteness results from the fact that there exist RGI relations among dimensional couplings that guarantee the vanishing of all beta-functions in certain N = 1 GUTs even to all orders. Additional developments in the soft supersymmetry breaking sector of N = 1 GUTs and FUTs lead to exact RGI relations, i.e. reduction of couplings, in this dimensionful sector of the theory, too. Based on the above theoretical framework phenomenologically consistent FUTs have been constructed. Here we review two FUT models based on the SU(5) gauge group. Confronting their predictions with the top and bottom quark masses and other experimental constraints a light Higgs-boson mass in the range M _H ～ 121–126 GeV has been predicted, in striking agreement with the recent experimental results from ATLAS and CMS. Furthermore naturally a relatively heavy s-spectrum emerged with coloured supersymmetric particles above ～1.5 TeV in agreement with the non-observation of those particles at the LHC. Restricting further the parameter space of the best version of the SU(5) FUT according to the reported accuracy of the Higgs boson mass and B-physics observables we find predictions for the rest of the Higgs masses and the s-spectrum.     

Experimental Consequences of the Asymptotically Free SU(5)-Model

The experimental consequences of the asymptotically free SU(5)-theory are presented which make it possible to judge about its reliability. The phenomenology of the τ-lepton is analysed with special attention paid to the so-called exotic processes which must solve whether “light” or “heavy” the ν_τ-neutrino should be. These exotic processes are completely forbidden within standard SU(5) schemes. The problem of the finite mass of the e, μ-neutrino which is a four-component Dirac spinor interacting via the Zel'dovitch-Mahmound-Konopinski scheme is discussed. The finiteness of such neutrino mass is a peculiarity of the SU(5) theory under discussion, although its value is not fixed here in a unique way. The version of the theory wherein m_ν ≦ 10⁻³ eV is of especial interest since in this case the proton instability and the finite e, μ-neutrino mass are found to be explained in a common way as manifestations of the existence of superheavy leptoquark bosons in the theory. The possibility is pointed out to explain within the SU(5) theory under consideration the existing experimental situations concerning the search for parity violating effects in atoms. It is shown that the present SU(5) theory encounters no difficulty in this respect and can recommend itself to any experimental set of data, whereas the standard schemes of SU(5) unification cannot. Cosmological consequences of the asymptotically free SU(5) theory are briefly discussed and the latter is found to rather reasonably explain a number of facts about the very early evolution of the Universe.     

On the Prasad-Sommerfield dyon (monopole) solution

We prove that the Prasad-Sommerfield dyon (monopole) solution for an SU(2) Yang-Mills field coupled with an SU(2) Higgs multiplet can be associated to a certain minimal immersion in S ³ (SU(2)) i.e. it has a differential-geometric content similar to that of self-dual solutions for the pure SU(2) Yang-Mills field. Implications of this result as well as possibilities to extend it to higher gauge groups are discussed.     

Heavy Quark Hadron Mass Scale

Without the spin interactions the hadron masses within a multiplet are degenerate. The light quark hadron degenerate multiplet mass spectrum is extended from the 3 quark ground state multiplets at J^P = 0⁻, ½⁺, 1⁻ to include the excited states which follow the spinorial decomposition of SU(2) × SU(2). The mass scales for the 4, 5, 6,… quark hadrons are obtained from the degenerate multiplet mass m₀/M = n²/α with n = 4, 5, 6,… The 4, 5, 6,… quark hadron degenerate multiplet masses follow by splitting of the heavy quark mass scales according to the spinorial decomposition of SU(2) × SU(2).     

Phase portrait ofSU(5) model. II intermediate phasesSU(3)×[U(1)]2 and [SU(2)]2×[U(1)]2

The effective potential of the scalar field in theSU(5) model has extrema with symmetry:SU(5),SU(4)×U(1),SU(3)×SU(2)×U(1),SU(3)×[U(1)]², [SU(2)]²×[U(1)]². In our recent paper it was shown that theSU(4)×U(1) phase as well asSU(3)×SU(2)×U(1) phase were stable at the nonzero temperature in a vast region of parameters. In the present paper it is found that the [SU(2)]²×[U(1)]² symmetric vacuum is unstable and theSU(3)×[U(1)]² symmetric vacuum can be metastable in the certain interval of the temperature. Domains of the three phases:SU(4)×U(1),SU(3)×SU(2)×U(1),SU(3)×[U(1)]²-could co-exist in the early. Universe.