Proof of Chiral Symmetry Breaking and Persistent Mass Condition in Vector- Like Gauge Theory

We give the details of rigorous proof of some mass inequalitiee in a vector-like gauge theory based on any simple group G. These mass inequalitiee lead to the conclusions that in auch a theory the chiral symmetries associated with all G representations of quarks must be spontaneously broken, and the persistent mass condition ie justified for any composite particles when the vacuum angle is zero or massless quarks exist.     

Parity violating vacuum currents on the random lattice

The Dirac operator is studied both with and without background gauge fieldson a three-dimensional random lattice. It is shown that the correct continuum limit for free fermions is recovered in the zero field case and the doubling problem evaded at tree level, the doubles acquiring masses of order the cut-off and decoupling from long-distance physics. The vacuum current is calculated in the presence of a nontrivial U(1) field and shown to be in agreement with continuum calculations.     

AN SU(6) GRAND UNIFIED MODEL

A model of grand unified theory based on SU(6) gauge group is proposed. It can accommodate two generations of ordinary fermions with V－A weak coupling and two generations of anomalous fermions with V+A weak coupling. In this model a new discrete symmetry is introduced which insures existence of fermions with lower masses when SU(6) gauge symmetry is spontaneously broken. We choose simple Higgs fields with appropriate vacuum expectation values so that the masses of anomalous fermions are heavier than those of ordinary fermions. This model also gives the same value of Weinberg angle, sin²θw=3/8, as in the usual SU(5) grand unified model at the grand unified scale.     

A Comment on the Light-Cone Vacuum in 1+1 Dimensional Super-Yang–Mills Theory

The discrete light-cone quantization (DLCQ) of a supersymmetric gauge theory in 1+1 dimensions is discussed, with particular attention given to the inclusion of the gauge zero mode. Interestingly, the notorious zero-mode problem is now tractable because of special supersymmetric cancellations. In particular, we show that anomalous zero-mode contributions to the currents are absent, in contrast to what is observed in the nonsupersymmetric case. An analysis of the vacuum structure is provided by deriving the effective quantum mechanical Hamiltonian of the gauge zero mode. It is shown that the inclusion of the zero modes of the adjoint scalars and fermions is crucial for probing the phase properties of the vacua. We find that the ground-state energy is zero and thus consistent with unbroken supersymmetry and conclude that the light-cone Fock vacuum is unchanged with or without the presence of matter fields.     

Vacuum alignment in supersymmetric gauge theories

The vacuum alignment problem is analyzed in the context of supersymmetric gauge theories with dynamical symmetry breaking. Three cases are distinguished, depending on whether the vacuum expectation value of the weak gauge current superfield in vacuum characterized by the orientation Ω is zero for all Ω, for some Ω, or for no Ω. In the first case, a non-renormalization theorem is proved to all orders in the weak coupling, showing that the usual criterion of minimizing the vacuum energy density is insufficient to determine the alignment, and possible resolutions of the problem are discussed. The second case is similar, except that the possible alignments are resricted to the range of Ω giving a vanishing VEV and the weak gauge group may then be broken non-minimally. In the third case, supersymmetry is itself broken at the tree level by the weak interactions. The supersymmetric generalization of the Schwinger mechanism for dynamical mass generation is described.     

Phases of renormalized lattice gauge theories with fermions

Starting from the formulation of gauge theories on a lattice we derive renormalization group transformation of the Migdal-Kadanoff type in the presence of fermions. We consider the effect of the fermion vacuum polarization on the gauge Lagrangian but we neglect fermion mass renormalization. We work out the weak coupling and strong coupling expansion in the same framework. Asymptotic freedom is recovered for the non-Abelian case provided the number of fermion multiplets is lower than a critical number. Fixed points are determined both for the U(1) and SU(2) case. We determine the renormalized trajectories and the phases of the theory.     

Composite Higgs scalars

We construct models in which the Higgs doublet whose vacuum expectation breaks SU(2) × U(10 is a bound state of massive strongly interacting fermions. The couplings of the composite Higgs to ordinary fermions are induced by heavy gauge boson exchange in the manner of extended technicolor. Other heavy gauge bosons generate a negative mass term for the Higgs.     

Vectorial versus axial Goldstone bosons

The Yukawa interactions of fermions with Goldstone bosons are given in closed form for an arbitrary renormalizable field theory to all orders of perturbation theory or for a general effective Lagrangian. Although the diagonal couplings are always pseudoscalar there is an important difference between spontaneously broken vector and axial-vector global symmetries. Compared to the axial case, the diagonal couplings of “vectorial” Goldstone bosons to charged fermions are suppressed by mixing angles or appear only via radiative corrections involving gauge fields. This general result may be relevant for the problem of flavour symmetry breaking in composite models.     

Top Quark,Heavy Fermions and the Composite Higgs Boson

We study the properties of heavy fermions in the vector-like representation of the electroweak gauge group SU(2)_W×U(1)_Y with Yukawa couplings to the standard model Higgs boson. Applying the renormalization group analysis, we discuss the effects of heavy fermions to the vacuum stability bound and the triviality bound on the mass of the Higgs boson. We also discuss the interesting possibility that the Higgs particle is composed of the top quark and heavy fermions. The bound on the composite Higgs mass is estimated using the method of Bardeen, Hill and Lindner (Phys. Rev. D 41 (1990) 1647), 150 GeV ≤ m_H ≤ 450 GeV.     

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^?.     

On the cancellation of fermion loops in chiral gauge theories in (1+1) dimensions

We show that for chiral gauge theories in (1+1) dimensions, with the gauge field coupling only to left(right)-handed fermions, the fermion loop contributions cancel in the vacuum expectation value of gauge invariant quantities, if the regularization of the theory respects the left(right)-handed coupling.     

Anomalous commutators in two-dimensional chiral gauge theories

It is shown that in two-dimensional models of chiral fermions coupled with background gauge fields, anomalous commutators of currents as well as of generators of time-independent gauge transformations are completely determined by the anomaly if the gauge group is semisimple. The four-dimensional case is commented on.     

Non-orientable foam and an effective planck mass for point-like fermions

A path-integral formalism is used to describe the propagation of fermions on spacetime foam in which virtual non-orientable handles are present. Because 2-component spinors cannot be defined on a non-orientable background, non-orientability is incompatible with chiral symmetry. Every fermion must have a partner (mirror fermion) with opposite chirality, that couples in the same way to the gauge bosons. Fermions appear to acquire an effective mass of the order of the Planck mass from propagation through non-orientable factors of the topologyy (a mechanism first suggested by Zel'dovich). A lower limit on the intrinsic size of leptons and quarks is apparently required to avoid excessively large masses. If one includes in the path integral a sum over all spinor structures, the effective mass arising from this mechanism is suppressed, but mirror fermions are still required. If one replaces the usual definition of parity by CP and assumes that only CP-reversing handles occur, and if, in addition, one sums over spinor structures, one can avoid both mirror fermions and a Planck-size effective mass for point fermions. However, in the context of a Kaluza-Klein theory, the restriction to CP-reversing handles is unnatural: both CP- and P-reversing handles can occur as vacuum fluctuations.     

Effect of bare mass on the Hosotani mechanism

It is pointed out that the existence of bare mass terms for matter fields changes gauge symmetry patterns through the Hosotani mechanism. As a demonstration, we study an SU(2) gauge model with massive adjoint fermions defined on M⁴S¹. It turns out that the vacuum structure changes at certain critical values of mL, where m (L) stands for the bare mass (the circumference of S¹). The gauge symmetry breaking patterns are different from models with massless adjoint fermions. We also consider a supersymmmetric SU(2) gauge model with adjoint hypermultiplets, in which the supersymmetry is broken by bare mass terms for the gaugino and squark fields instead of the Scherk–Schwarz mechanism.     

Proof of chiral symmetry breaking in strongly coupled lattice gauge theory

We study chiral symmetry in the strong coupling limit of lattice gauge theory with staggered fermions and show rigorously that chiral symmetry is broken spontaneously in massless QED and the gauge-invariant Nambu-Jona-Lasinio model if the dimension of spacetime is at least four. The results for the chiral condensate as a function of the mass imply that the mean-field approximation is an upper bound for this observable which becomes exact as the dimension goes to infinity. For the model with gauge groupU(N),N=2,3,4, we prove that chiral long-range order exists at zero mass in four or more dimensions. Address after August 1991: Mathematics Department, University of British Columbia, Vancouver, Canada V6T1Y4     

Constraints on Masses of Composite Particles Containing Exotic Quarks

It is shown based on some rigorously proven mass inequalities that some of the composite particles containing the SU_c(3) 6-plet and 8-plet quarks (q₆ and q₈) could appear in TeV physics if q₆ and q₈ exist in nature and can be used in the scheme of dynamical symmetry breaking of the electroweak gauge group.     

The role of surface variables in the vacuum structure of Yang-Mills theory

The structure of the vacuum in the SU(2) Yang-Mills theory is discussed for general gauges. The original discussions given by Callan et al. and Jackiw and Rebbi are restricted to a particular class of gauge conditions. We show that the periodic vacua and the “vacuum seizing” in the presence of massless fermions can be realized in any gauge by recognizing the independent dynamical role of surface variables defined at spatial infinity.     

Charged condensation

We consider Bose–Einstein condensation of massive electrically charged scalars in a uniform background of charged fermions. We focus on the case when the scalar condensate screens the background charge, while the net charge of the system resides on its boundary surface. A distinctive signature of this substance is that the photon acquires a Lorentz-violating mass in the bulk of the condensate. Due to this mass, the transverse and longitudinal gauge modes propagate with different group velocities. We give qualitative arguments that at high enough densities and low temperatures a charged system of electrons and helium-4 nuclei, if held together by laboratory devices or by force of gravity, can form such a substance. We briefly discuss possible manifestations of the charged condensate in compact astrophysical objects.     

Composite massless fermions in strongly coupled lattice gauge theories

We show that a class of strongly coupled lattice gauge theories with fermions in real representations of the gauge group do not have chiral symmetry breaking. The resulting spectrum of massless composite fermions satisfies 't Hooft's constraints if the model is naively extrapolated to the continuum limit. We argue that it is in fact the correct spectrum of the continuum gauge theory.     

The fermion determinant in two dimensional Minkowski space: Zeros and related properties

We present a detailed analysis of the non-abelian determinant of massless fermions in two dimensional Minkowski space. In the framework of the external field problem, the determinant vanishes if the out- and ingoing vacua are orthogonal; the gauge potentials for which this happens are identified. Causality implies that the effective action obtained from the sum of fermion loops has the right singularity at a zero of the determinant. Such a zero can be reached by a continuous deformation of a potential with non-vanishing determinant. The set of zeros exhibits a rich structure.Work partially supported by the Swiss National Science Foundation