Does the 2d Higgs-yukawa Model Have a Symmetric Phase at Small Yukawa Coupling Region?

We show that at arbitrary value of the scalar self coupling and small Yukawa coupling y the 2d Higgs-Yukawa model with Z(2) symmetry remains in the broken phase and the model is asymptotically free: y → 0 as the cut-off Λ → ∞. This is in agreement with a recent conjecture based on numerical simulation results. The Higgs-Yukawa model has been shown to be equivalent to a generalized Nambu-JonaLasinio (NJL) model in 2 < d ≤ 4 dimension [1, 2]. However, much less is known for its properties in d = 2. The lattice action for the Higgs-Yukawa model with a Z(2) symmetry is given by S = Sh[φ] + ∑ x,z ψ̄i(x)M(x, z)ψi(z) , i = 1, 2, ..., Nf , (1) where the scalar action Sh[φ] has the form Sh[φ] = −2κ ∑