Ising-Gauge Ising-Higgs System in Two Dimensions

Abstract

A brief review for Refs.[1] and [2] is presented. -gauge and -Higgs action on 2-dimensional lattice is given in terms of standard procedures. The duality transformations for lattice gauge fields are used for relating such model to Ising model with . We regard the close to the critical point of 2-dimensional Ising model as a concrete realization of the renormalized, continuous and Euclidean invariant (i.e., Lorentz invariant) gauge field theory, or Ising-gauge Ising-Higgs field theory in two dimensions. When or , respectively, while , and let as well as be finite, both the symmetry non-breaking -gauge field theory and the spontaneous breaking -gauge field theory are obtained. Then, we use recent calculation of correlation functions of 2-dimensional Ising model in both the absence and the presence of a magnetic field to study this -gauge -Higgs system. The correlations (i.e., renormalized Green functions) of two plaquettes and the gauge-invariant correlations of Higgs field are investigated in various cases. Comparing with 4-dimensional QCD, and can be understood as the glueball-correlations and the meson-correlations in this model, respectively. Their isolated poles in momentum space correspond to the bound states, and the poles' locations give mass spectrums of these states. For pure -gauge theory (i.e., ), if the gauge symmetry does not break ( case), has a leading pole, on the contrary, if the gauge symmetry is spontaneous breaking ( case), has a leading cut. In the presence of Higgs matter fields (i.e., ) the leading singularity of still is a pole, however, 's cut is broken into a series of poles of . On the other hand, as the gauge fields are “frozen” has a two-particle cut (and no single particle pole). But in the presence of -gauge fields, the Higgs fields will be confined, that is, has a leading pole and has a series of poles. After a tedious calculation, we confirm that the 's pole is a particle with determinate mass indeed. All these analytic calculations and expressions tell us the physics contents of 2-dimensional Ising-gauge Ising-Higgs system and show the criteria for confinement in the presence of matter. When the -gauge symmetry does not break, we may say the confining potential is very strong. However, as the -gauge symmetry breaks down spontaneously, the confining potential becomes very weak. The dynamical Higgs field imparts the dynamics to the renormalized gauge field correlations, which makes the leading cut of break into a series of poles of . It seems to be very remarkable that strong interaction and weak interaction now appear in one unification gauge theory formalism in which the confinement of elementary excitations into “mesons” or “glueballs” comes from the explicit nonperturbation calculations.