  The vacuum polarization due to chiral fermions on a 4--dimensional Euclidean
lattice is calculated according to the overlap prescription. The fermions are
coupled to weak and slowly varying background gauge and Higgs fields, and the
polarization tensor is given by second order perturbation theory. In this order
the overlap constitutes a gauge invariant regularization of the fermion vacuum
amplitude. Its low energy -- long wavelength behaviour can be computed
explicitly and we verify that it coincides with the Feynman graph result
obtainable, for example, by dimensional regularization of continuum gauge
theory. In particular, the Standard Model Callan--Symanzik RG functions are
recovered. Moreover, there are no residual lattice artefacts such as a
dependence on Wilson--type mass parameters.
