Perturbation Theory at Finite Extent of Fifth Dimension for Vacuum Overlap Formula of Chiral Determinant -- Continuum Limit Case --

Taking into account of the boundary condition in the fifth direction which is derived from the lattice Wilson fermion, we develop a theory of five-dimensional fermion with kink-like and homogeneous masses in finite extent of the fifth dimension. The boundary state wave functions are constructed explicitly and the would-be vacuum overlap is expanded by using the propagator of the theory. The subtraction is performed unambiguously at the finite extent with the help of the dimensional regularization. Then the limit of the infinite extent is evaluated. The consistent anomaly in four dimensional theory is finitely obtained. Each contribution to the vacuum polarization is vector-like. It is the lack of the massless mode in the fermion with negative homogeneous mass that leads to the correct chiral normalization. Gauge noninvariant piece remains due to the breaking of the boundary condition by the dimensional regularization.