  We study the phase structure of SU(2) gauge-Higgs model in two dimensions
using lattice simulations. We show the result for the plaquette expectation
value, static potential, and W propagator. Our results suggest that a
confinement-like region and a Higgs-like region appear even in two dimensions.
The behavior of the plaquette expectation value is consistent with a smooth
cross-over in accordance with the Fradkin-Shenker-Osterwalder-Seiler theorem.
In the confinement-like region, the static potential seems to rise linearly
with string breaking at large distances, while in the Higgs-like region there
seems to be a massive behavior which means that the BEH mechanism occurs. The
correlation length obtained from the W propagator has a finite maximum between
these phases, which supports no second-order phase transition. Based on these
results, we suggest that there is no phase transition in two dimensions.
