Fermion Number Violation and a Two-dimensional Higgs Model

The investigation of topological properties of the gauge field in a two-dimensional Higgs model can help in understanding anomalous fermion number violation. 1. A simple model for fermion number violation Fermion number, which is the sum of baryon number and lepton number (B+L), is not conserved in the Standard Model 1. This is due to the anomaly in the fermion current. The lattice formulation of the anomalous fermion number non-conservation is problematic 2 , because it has to do with the chiral SU(2) L ⊗U(1) Y gauge couplings and, as is well known, there is a difficulty with chiral gauge theories on the lattice (see, for instance, the reviews 3,4). There is, however, an approximation of the electroweak sector of the standard model which can be studied with standard lattice techniques, namely the limit when the SU(3) colour ⊗ U(1) hypercharge gauge couplings are zero 5,6. A simple prototype model is the standard SU(2) L Higgs model coupled to an even number 2N f of fermion doublets. One can take, for simplicity, N f = 1 but the extension to N f > 1 is straightforward. The lattice action can be written in terms of the fermion doublet fields ψ (1,2)x with an off-diagonal Majorana mass and Majorana-like Wilson term. It is, however, technically more convenient to consider a Dirac-like form with ψ ≡ ψ 1 and the mirror fermion field χ defined by χ x ≡ ǫ −1 Cψ T 2x , χ x ≡ ψ T 2x ǫC. (1) with the charge conjugation matrix C and ǫ acting in isospin space. In terms of ψ and χ one obtains the mirror fermion action for chiral gauge fields, which is well suited for studying the physically relevant phase with broken symmetry. Before doing numerical simulations in the four dimensional SU(2) L gauge model for anomalous fermion number violation, it is useful to study a simple U(1) toy model in two dimensions, which has often been studied in this context (see e. g. 7). The lattice action depending on the compact U(1) gauge field U xµ = exp(iA µ (x)), (µ = 1, 2) and, for simplicity, fixed length Higgs scalar field φ(x), |φ(x)| = 1 can be