  As a first step towards a nonperturbative investigation of the gauge-fixing
(Rome) approach to lattice chiral gauge theories we study a U(1) model with an
action that includes a local gauge-fixing term and a mass counterterm for the
gauge fields. The model is studied on the trivial orbit so that only the
dynamics of the longitudinal gauge degrees of freedom is taken into account.
The phase diagram of this higher-derivative scalar field theory is determined,
both in the mean-field approximation and numerically. The continuum limit of
the model corresponds to a continuous phase transition between a ferromagnetic
(FM) phase where the global U(1) symmetry is broken, and a so-called helicoidal
ferromagnetic (FMD) phase with broken U(1) symmetry and a nonvanishing
condensate of the vector field. The global U(1) symmetry is restored in this
continuum limit. We show that our data for the magnetization in the FM and FMD
phases are in good agreement with perturbation theory.
