The vacuum polarization due to chiral fermions on a 4-dimensional Eucl
idean 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 per
turbation theory. In this order the overlap constitutes a gauge-invari
ant regularization of the fermion vacuum amplitude. Its low-energy-lon
g-wavelength behaviour can be computed explicitly and we verify that i
t coincides with the Feynman graph result obtainable, for example, by
dimensional regularization of continuum gauge theory. In particular, t
he Standard Model Callan-Symanzik RG functions are recovered. Moreover
, there are no residual lattice artefacts such as a dependence on Wils
on-type mass parameters.