Grand unification of effective gauge theories

An effective non-renormalizable SU(3)×SU(2)×U(1) invariant gauge theory results at ordinary energies when superheavy fields are integrated out from a grand unified theory based on a simple gauge group G. The solutions of the second-order renormalization-group equations for the gauge coupling constants of the effective theory are examined. General formulae for the superheavy vector boson mass and for sin²θ near M_W are given in this approach to grand unification. The superheavy vector boson mass is plotted against the QCD scale parameter Λ for a certain set of grand unified models. Corrections to the prediction when the set of models is enlarged are discussed, and illustrated with examples from G≡SU(5) and O(10).     

Grand unification at the subcomponent level

The Casalbuoni-Gatto subcomponent model of quarks and leptons is generalized allowing for an arbitrary number of subcomponents. It is shown that there are only a limited number of cases where the subcolour can be embedded in a semi-simple grand unification scheme. The most interesting models lead to an SU(7) ? SU(7) grand unification at the subcomponent level. In one of them there is also a natural place for a hypercolour (technicolour) group SU(2)_hc.     

On the phenomenological group in the unified SO(10) model

We analyze possible low energy effects of the additional U(1) contained in the symmetry breaking chain: SO(10)→SU(5) ? U(1) →…, stressing the importance of considering extensions of SU(2) ? U(1) as subgroups of grand unified models, in order to use the relations between the coupling constants provided by the renormalization group. We also investigate the possibility of employing this extra U(1) for an explanation of the possible discrepancy between the experimental value of sin²θ_w and its renormalized value in the SU(5) model.     

On the structure of the low energy superpotential in superstring theory

We show that after the gauge symmetry breaking of the E₆ grand unified group to the H=SU(3)^c×SU(2)_L×U(1)_Y× U(1)_E subgroup by the Hosotany mechanism a number of additional Yukawa terms not present in a decomposition of 27×27×27 may appear in the low energy superpotential. Some of these terms cause a rapid proton decay.     

Gauge hierarchies of SU(N) grand unification

The structure of spontaneous breaking of SU(N) gauge symmetry for grand unification is investigated. The results obtained are applied to the analysis of SU(8) symmetry for which possible ways of breaking and intermediate symmetries are considered. It is assumed that the SU(8) group unifies the subgroups of colour, standard electroweak and horizontal symmetries. We find conditions which it is necessary to impose on the vacuum expectation values of Higgs multiplets to provide an arbitrary breaking pattern of SU(N) symmetry and conserve any intermediate symmetry. If in the SU(8) models considered fermions and mirror fermions do not violate the (V-A) and (V+A) structure of weak interactions, then their masses should not be greater than ～10² GeV. It is also shown that the contributions of fermion and Higgs multiplets to the renormalization group equation for the coupling constant of any subgroup of SU(N) are identical. Renormalization group identities for the case of arbitrary SU(N) breaking are given where the contribution of Higgs multiplets have been taken into account (but they cancel each other). Using these identities one can calculate the mass values for the breaking of the intermediate symmetries in the SU(8) models, and also exclude part of the possible breaking patterns.     

Early unification and the Weinberg angle in the SU(4)4 grand unified model

In the framework of the grand unified gauge group SU(4)⁴ we discuss possibilities to reconcile the low unification scale (10⁵?10⁷ GeV) with the acceptable value of sin²θ_w. We consider various specific models which differ by the values of the intermediate mass scale, the choice of the fermion multiplets and by the embedding of the electroweak group SU(2) into SU(4)⁴. The class of theories with early unification and correct sin²θ_w is constructed. They all predict new non-sequential fermions which are SU(2)_L,R singlets and have unconventional electric charges. Cosmological implications of such theories are discussed and it is argued that new particles may well account for the positive results of searches for fractional charge in terrestrial matter.     

SU(5) Higgs problem with adjoint + vector representations

Recently Kim has given a general method, using group-invariant orbit parameters, for determining the energy and residual symmetry of the Higgs potential minimum. In this paper we illustrate the method by working out the case of a quartic SU(5) Higgs potential with $\text{5}$ and $\text{24}$ higgsons. In this method the Gell-Mann-Slansky conjecture concerning possible little groups of the potential minimum takes a geometric form, which is verified for our case. The results are used to discuss the hierarchical symmetry breaking of SU(5) grand unification theory. We generalize our results to the SU(N) adjoint + vector models, which are all closely related.     

SU(8) grand unification

A grand unified gauge theory based on the group SU(8) is presented. The set of rules motivating the choice of SU(8) is given. The particle content of the theory suggests a natural embedding of a horizontal SU(3) symmetry in the SU(8) grand unified symmetry.     

Grand unification with local supersymmetry

We study general conditions for obtaining spontaneous breaking of local supersymmetry in N = 1 supergravity coupled to supersymmetric matter. We consider in particular the coupling of N = 1 supergravity to grand unified theories like SU(5) and study the conditions which must be met in order to obtain a realistic model. Specific models are built in which local supersymmetry is broken at a scale √M_Wm_p ～ 10¹⁰ GeV. This breaking of supersymmetry is only detected at low energies through soft terms breaking explicitly the global supersymmetry. These soft terms (scalar masses, gaugino masses and trilinear scalar couplings) are renormalized at low energies according to the renormalization group. The (mass)² of the Higgs doublet evolve towards negative values at low energies giving rise to SU(2) × U(1) breaking as a radiative effect of local supersymmetry breaking. We finally point out the possible relevance of non-renormalizable superpotentials for the problem of fermion masses.     

On the addition of vector-like quarks to the standard model

We analyze some of the implications of adding vector-like isosinglet quarks to the standard model and its simplest extension based on the low-energy gauge group SU(3)_c × SU(2) × U(1)^m, which naturally arises in some grand unified theories as well as in some versions of superstring theories. Some of the novel features of this class of models are pointed out: non-unitarity of the Cabibbo-Koyabashi-Maskawa matrix, new CP-violating phases, flavour-changing neutral currents. We derive the CP invariance restrictions on the various quark mass terms and propose a parametrization of the quark mixing matrix which is particularly suitable for models with vector-like quarks. Constraints on these models are derived from rare kaon decays and the value of the K_L − K_S mass difference.     

Fundamental fermion representations in generalisations of the SU(5) grand unified gauge theory

Generalisations of the SU(5) grand unified gauge theory are discussed. It is assumed that the gauge group is simple, and that the theory is both anomaly free and asymptotically free. All possible fundamental fermion representations are determined given that the fermions are massless at the unification level, but acquire mass at the level of the exact SU(3) × U(1) symmetry. No a priori restriction to standard colour SU(3) representations is made. It is found that E₆, SO(10) and SU(n) with n ? 5 are the only acceptable gauge groups. Standard colour solutions are legion, but dull, incorporating at the SU(5) level, nothing other than p generations or copies of the familiar representation $\text{10}\text{+}\text{5}$. Exotic colour solutions are sparse, but interesting. Nine of these, all associated with SO(10), can accomodate those quarks and leptons currently thought to be fundamental, along with such things as colour sextets and octets, as well as doubly charged leptons.     

Quantum fluctuations in the vicinity of an instanton in the SU(N) group

A general form of the quadratic Lagrangian in the presence of a self-dual classical field has been obtained. The calculation of the amplitude of the vacuum-vacuum transition in the SU(N) group has been reduced to calculation of similar amplitudes in the SU(2) group for scalar particles with different isospins. The value of the amplitude for one instanton has been obtained.     

Threshold effects in SUSY and non-SUSY GUTs

We discuss recent contributions on threshold effects in grand unfiied theories including minimal SUSY SU (5), non-SUSY modifications of the grand desert in SU(5) and SO(10), and SO(10) with single intermediate symmetires. Consequences of theorems on vanishing GUT-scale corrections to sin² θ_w in SO(10) with SU(2) _L XSU(2) _R XSU(4) _c (g_2l =g_2R ) intermediate symmetry are discussed and vanishing corrections on the inter-mediate scale are explicitly demonstrated where predictions are more precise. Threshold and higher dimensional operator effects in SUSY SU(5) recently derived by a number of authors are presented.     

Mixing of leptonic number in a general SU(2)L ⊗ U(1) ⊗ SU(2)R model

We discuss a general leptonic model based on the SU(2)_L ? U(1) ? SU(2)_R gauge group and we present general results for the ? → ?′ γ and ν? → ν?′γ processes, which can be used for any model based on the SU(2)_L ? U(1) ? SU(2)_R gauge group or on any of its subgroups. The predictions of the μ → eγ branching ratio and of the neutrino lifetime for various models discussed in the literature are reviewed. The main results are summarized in tables.     

Left-right symmetry breaking scale and supersymmetry in unified theories

We propose a class of supersymmetric grand unified models where parity and SU(2)_R breaking scales are widely separated and compatible with a low-lying mass for the right-handed gauge boson W_R. The intermediate symmetry SU(4)_c×SU(2)_L×SU(2)_R and Higgs content are uniquely fixed if m_WR < 10⁹ GeV. The unification scale lies within an order of magnitude below the Planck mass.     

The Majoron and left-handed neutrino masses in SU(5)

We consider the spontaneous breaking of B?L in SU(5) in connection with left-handed neutrino masses. We show that the ensuing Goldstone boson, “the majoron”, is harmless. The low-energy predictions are the same as in the SU(2)_L × U(1) majoron model which, in particular, predicts the existence of doubly and singly charged scalars, with masses ?250 GeV and with a mass ratio of √2. Going to SU(5), one would expect these scalars to escape to the grand unified mass. This does not happen and, surprisingly, even the ratio √2 is preserved.     

Grand unified model with a stable proton and no axion problem

We present a grand unified model of the strong, electromagnetic and weak interactions based on a local SU(8)_L×SU(8)_R gauge theory which possesses a global U(8)_L × U(8)_R invariance. We break the symmetry down to the standard SU(3)_C × SU(2)_L × U(1) model, with the proton remaining stable and the left-handed neutrinos obliged to remain massless. A novel feature of our model is the simultaneous absence of both strong CP violations and of axions.     

Symmetric versus antisymmetric mass matrices in grand unified theories

We point out some interesting consequences of antisymmetric fermionic mass terms in grand unified theories, which follow from the symmetry properties of the Yukawa-type fermion-Higgs interactions. In SU(5), we show that an antisymmetric mass matrix M(u) in the up-quark sector arises when the Higgs 5 is replaced by 45. IOn SO(10), all the fermion sectors are characterized by such mass matrices if the neutrinos are required not to pick up the ordinary fermionic mass scale. In the recently proposed vertical-horizontal symmetric SU(5) × SU(5) scheme, a fully antisymmetric M(u) is naturally accompanied by exactly five zeroth-order massless neutrinos.     

A solution to the strong CP problem in the SU(5) model

The possibility of naturally suppressing strong CP violation in the grand unified SU(5) model is examined. We show that a recent solution to the quark-lepton mass ratio problem by Georgi and Jarlskog can be extended in such a way as to yield a zero strong CP-violating parameter θ at the tree level and a small value for it coming from higher-order effects, along with reasonable mass ratios and mixing angles. Several phenomenological implications of the model are noted.     

Can the SU(5) running be neglected in the minimal N = 1 Sugra model?

The radiative breaking of the electroweak symmetry is analysed in the minimal N = 1 supergravity model when the running of the parameters between the Planck and the grand unification scales is considered. The SU(5) running opens up the parameter space of the SU(2) × U(1) breaking solutions, and in particular all the values ߊ₀| > 3 are allowed. The SU(5) running will modify the structure of the low-energy mass spectra and will lead to the possibility of having realistic ones for very small values of the gravitino mass m_3/2. Indeed, if the SU(5) evolution is considered, no lower bound on m_3/2 is needed to obtain acceptable spectra.