Exotic phases of finite temperature gauge theories

We calculate the phase diagrams at high temperature of SU(N) gauge theories with massive fermions by minimizing the one-loop effective potential. Considering fermions in the adjoint (Adj) representation at various N we observe a variety of phases when N_f2 Majorana flavours and periodic boundary conditions are applied to fermions. Also the confined phase is perturbatively accessible. For N=3, we add Fundamental (F) representation fermions with antiperiodic boundary conditions to adjoint QCD to show how the Z(3)-symmetry breaks in the confined phase.     

There is No “Theory of Everything” Inside E8

We analyze certain subgroups of real and complex forms of the Lie group E₈, and deduce that any “Theory of Everything” obtained by embedding the gauge groups of gravity and the Standard Model into a real or complex form of E₈ lacks certain representation-theoretic properties required by physical reality. The arguments themselves amount to representation theory of Lie algebras in the spirit of Dynkin’s classic papers and are written for mathematicians.     

Low-energy neutral current phenomenology and grand unified theories

We derive necessary and sufficient conditions to be satisfied by any expanded electroweak gauge model in order to reproduce the standard model low-energy neutral current predictions. These conditions imply several constraints on the neutral gauge boson masses and quantum number assignments for the ordinary fermions. Using these conditions, we prove that the popular grand unified theories based on the gauge groups SO(10) and E₆ can only accomodate trivial extensions of the standard model. As a consequences, if any of these grand unified models works at some energy scale, present low-energy neutral current phenomenology implies that the Z-boson must be produced with the expected mass and couplings to the ordinary fermions. Any additional neutral gauge boson (with the possible exception of very heavy ones) could only be produced in hadronic collisions and it would not decay in e⁺e^?.     

Superconformal Indices of $${{\mathcal N}=4}$$ SYM Field Theories

Superconformal indices (SCIs) of 4d N = 4{{\mathcal N} = 4} SYM theories with simple gauge groups are described in terms of elliptic hypergeometric integrals. For F ₄, E ₆, E ₇, E ₈ gauge groups this yields first examples of integrals of such type. S-duality transformation for G ₂ and F ₄ SCIs is equivalent to a change of integration variables. Equality of SCIs for SP(2N) and SO(2N + 1) group theories is proved in several important special cases. Reduction of SCIs to partition functions of 3d N=2{{\mathcal N}=2} SYM theories with one matter field in the adjoint representation is investigated, corresponding 3d dual partners are found, and some new related hyperbolic beta integrals are conjectured.     

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.     

Higgs boson production at CLIC in SUSY models with hierarchy of the vacuum expectation values

Many theoretical schemes predict the existence of unconventional quarks and leptons not fitting the standard pattern of fermion families. We consider two possible kinds of non-standard fermions, mirror fermions and exceptional fermions. Mirror fermions arise in a variety of models ranging from family unification to extended supersymmetry and Kaluza-Klein theories; exceptional fermions come along with the groupE ₆ which is believed to be the low energy gauge symmetry of the superstring theory. We discuss some physical properties of these non-standard particles relevant for the LEP     

Dynamical supersymmetry breaking and gauge anomalies

Some aspects of supersymmetric gauge theories and discussed. It is shown that dynamical supersymmetry breaking does not occur in supersymmetric QED in higher dimensions. The cancellation of both local (perturbative) and global (non-perturbative) gauge anomalies are also discussed in supersymmetric gauge theories. We argue that there is no dynamical supersymmetry breaking in higher dimensions in any supersymmetric gauge theories free of gauge anomalies. It is also shown that for supersymmetric gauge theories in higher dimensions with a compact connected simple gauge group, when the local anomaly-free condition is satisfied, there can be at most a possibleZ ₂ global gauge anomaly in extended supersymmetricSO(10) (or spin (10)) gauge theories inD=10 dimensions containing additional Weyl fermions in a spinor representation ofSO(10) (or spin (10)). In four dimensions with local anomaly-free condition satisfied, the only possible global gauge anomalies in supersymmetric gauge theories areZ ₂ global gauge anomalies for extended supersymmetricSP(2N) (N=rank) gauge theories containing additional Weyl fermions in a representation ofSP(2N) with an odd 2nd-order Dynkin index.     

Quarks and leptons

We review the physics of quarks and leptons within the framework of gauge theories for the weak and electromagnetic interactions. The Weinberg-Salam SU(2) × U(1) theory is used as a “reference point” but models based on larger gauge groups, especially SU(2)_L × SU(2)_R × U(1), are discussed. We distinguish among thre “generations” of fundamental fermions: The first generation (e^?, ν_e, u, d), the second generation (μ^?, ν_μ, c, s) and the third generation (τ^?, ν_τ, t, b). For each generation we discuss the classification of all fermions, the charged and neutral weak currents, possible right-handed currents, parity and CP-violation, fermion masses and Cabibbo-like angles and related problems. We review theoretical ideas as well as experimental evidence, emphasizing open theoretical problems and possible experimental tests. The possibility of unifying the weak, electromagnetic and strong interactions in a grand unification scheme is reviewed. The problems and their possible solutions are presented, generation by generation, but a brief subject-index (following the table of contents) enables the interested reader to follow any specific topic throughout the three generations.     

Embedding of non-simple Lie groups,coupling constant relations and non-uniqueness of models of unification

A general derivation of the coupling constant relations which result on embedding a non-simple group like SU _L (2) ⇔ U(1) in a larger simple group (or graded Lie group) is given. It is shown that such relations depend only on the requirement (i) that the multiplet of vector fields form an irreducible representation of the unifying algebra and (ii) the transformation properties of the fermions under SU _L (2). This point is illustrated in two ways, one by constructing two different unification groups containing the same fermions and therefore have same Weinberg angle; the other by putting different SU _L (2) structures on the same fermions and consequently have different Weinberg angles. In particular the value sin²ϑ=3/8 is characteristic of the sequential doublet models or models which invoke a large number of additional leptons likeE ₆, while addition of extra charged fermion singlets can reduce the value of sin² ϑ to 1/4. We point out that at the present time the models of grand unification are far from unique.     

On the theory of interacting fields in the Foldy-Wouthuysen representation

The overview is devoted to quantum electrodynamics (QED) and the Standard Model in the Foldy-Wouthuysen representation. The Hamiltonian H _FW in the form of a power series in charge e is obtained as applied to the electromagnetic interaction in the FW representation. Quantum electrodynamics in lowest-order perturbation theory is examined. Calculations of specific QED processes are presented. For external fermion lines (p _f ² = m ²), a possibility to expand the scattering matrix, in powers of the coupling constant with matrix elements, not including fermion propagators, is shown. To take into account particle-antiparticle interaction, a modification of the Foldy-Wouthuysen representation is proposed. Fermions in the modified FW representation can be in two states that are characterized by the sign of a third component of the isotopic spin T _f ³ . The sign of T _f ³ is connected with the sign at mass terms in the modified Hamiltonian H _FW. Real fermions (p _f ² = m _f ² ), as well as antifermions, can interact with each other, while real fermions with a given sign of T _f ³ can only interact with real antifermions with the opposite sign of T _f ³ , and vice versa. The formulation of the Standard Model in the FW representation does not necessarily require an interaction of Higgs bosons with fermions. In this approach, the role of Higgs bosons narrows considerably as they are responsible only for gauge invariance of the theory and interact only with gauge bosons. Quantum electrodynamics in the FW representation is invariant under C, P, and T transformations. Weak interaction does not conserve C and P parity, but conserves combined CP parity. The theory allows a connection of CP violation and total or partial violation of isotopic symmetry in the modified Foldy-Wouthuysen representation.     

The N = 4 supersymmetric E8 gauge theory and coset space dimensional reduction

Reasons are given to suggest that the N = 4 supersymmetric E₈ gauge theory be considered as a serious candidate for a physical theory. The symmetries of this theory are broken by a scheme based on coset space dimensional reduction. The resulting theory possesses four conventional generations of low-mass fermions together with their mirror particles.     

The small-volume expansion of gauge theories coupled to massless fermions

The small-volume expansion of the low-lying glueball states for SU(2) and SU(3) gauge theory, coupled to massless fermions with periodic and antiperiodic boundary conditions, is determined. For SU(3) with periodic boundary conditions the vacuum is eightfold degenerate and breaks part of the cubic group spontaneously. In all cases the scalar-to-tensor mass ratio m_A1⁺⁺/m_E⁺⁺ is 1.1 to 1.3 as in the pure-gauge case. We also discuss chiral symmetry.     

Boundaries forSU(3) c xU(1)el lattice gauge theory with a chemical potential

It is shown forSU(N) andU(1) gauge groups that periodic spatial boundary conditions, as commonly used in lattice simulations, are not possible in the charged sectors of a local gauge theory. For charge-conjugate (C-)periodic boundary conditions the effective gauge action of fermions is derived. For nonzero chemical potential, the breakdown of translational invariance induced by the breakdown ofC symmetry is discussed. If translational invariance is abandoned, (anti)periodic spatial b.c. for fermions and for theSU(3) gauge field andC-periodic b.c. for theU(1) gauge field can be used.     

Unification of gravity and elementary particle interactions in 26 dimensions?

The presence of an 11-cocycle in ten-dimensional theories containing an E₈×E₈ Yang-Mills theory coupled to supergravity suggests that the fundamental theory is a 26-dimensional string theory based on the Leech lattice. A theory with equal numbers of bosons and fermions is constructed by truncating a certain natural infinite-dimensional representation of the Monster sporadic group. This theory appears to be a certain combination of pieces of heterotic and N = 2 superstring.     

E 6,7,8 magnetized extra-dimensional models

We study 10D super Yang–Mills theory with the gauge groups E ₆, E ₇ and E ₈. We consider the torus/orbifold compactification with magnetic fluxes and Wilson lines. They lead to 4D interesting models with three families of quarks and leptons, whose profiles in extra dimensions are quasi-localized because of magnetic fluxes.     

Orbital quantization in a system of edge Dirac fermions in nanoperforated graphene

The dependence of the electric resistance R of nanoperforated graphene samples on the position of the Fermi level E _F, which is varied by the gate voltage V _g, has been studied. Nanoperforation has been performed by irradiating graphene samples on a Si/SiO₂ substrate by heavy (xenon) or light (helium) ions. A series of regular peaks have been revealed on the R(V _g) dependence at low temperatures in zero magnetic field. These peaks are attributed to the passage of E _F through an equidistant set of levels formed by orbitally quantized states of edge Dirac fermions rotating around each nanohole. The results are in agreement with the theory of edge states for massless Dirac fermions.     

Systematic study of horizontal gauge theories

We analyze all the possible continuous horizontal gauge groups G _H in relation with their possibility to explain m _b ? m _t. We assume that the only effective fermionic degrees of freedom correspond to the known fermions but allow the possibility of adding a right handed neutrino to each family. We assume that the Higgs fields which generate masses for these fermions, trough renormalizable Yukawa couplings, transform as an irreducible representation of SU(3)_c ? SU(2)_L ? U(1)_Y ? G_H. Under these assumptions we find two U(1)_H or U(1) _H1 ? U(1) _H2 models free of anomalies and able to guarantee that only the top has a renormalizable mass-generating Yukawa coupling.     

N = 1 Dualities for Exceptional Gauge Groups and Quantum Global Symmetries

We discuss our attempts to generalize the known examples of dualities in N = 1 supersymmetric gauge theories to exceptional gauge groups. We derive some dual pairs from known examples connected to exceptional groups and find an interesting phenomenon: sometimes the full global symmetry is “hidden” on the magnetic side. It is not realized as a symmetry on the fundamental fields in the Lagrangian. Rather, it emerges as a symmetry of the quantum theory. We then focus on an approach based on self-dual models. We construct duals for some very special matter content of E₇, E₆ and F₄. Again we find that the full global symmetry is not realized on the fundamental fields.     

Hierarchical fermion masses from grand unification

A mechanism is presented for generating fermion masses as a reflection of the superheavy fermions existing in a class of grand unified theories. By means of it we propose a way of understanding how, giving a driving mass term for the top-generation, the charm-generation mass results as a relative first-order perturbation in the unified gauge coupling g²/4π ≈ 10^?2, and the up-generation mass as a second-order perturbation. An illustrative calculation of the charm-generation mass is described in a model based on E₆.