  We propose a novel gauge-invariant regularization for the perturbative chiral
gauge theory.Our method consists of the two ingredients: use of the domain-wall
fermion to describe a chiral fermion with Pauli-Villars regulators and
application of the di- mensional regularization only to the gauge field. This
regularization is implemented in the Lagrangian level, unlike other
gauge-invariant regularizations (eg. the covariant regularizations). We show
that the Abelian (fermion number) anomaly is reproduced correctly in this
formulation. We also show that once we add the counter terms to the full
theory, then the renormalization in the chiral gauge theory is automatically
achieved.
