Lattice Chiral Gauge Theory with Finely-Grained Fermions

We discuss the problem of formulating the continuum limit of chiral gauge theories ($\chi$GT) in the absence of an explicitly gauge-invariant regulator for the fermions. A solution is proposed which is independent of the details of the regulator, wherein one considers two cutoff scales, $\Lambda_f >> \Lambda_b$, for the fermions and the gauge bosons respectively. Our recent non-perturbative lattice construction in which the fermions live on a finer lattice than do the gauge bosons, is seen to be an example of such a scheme, providing a finite algorithm for simulating $\chi$GT. The essential difference with previous (one-cutoff) lattice schemes is clarified: in our formulation the breakage of gauge invariance is small, $O(\Lambda^2_b/\Lambda^2_f)$, and vanishes in the continuum limit. Finally, we argue against 2-D models being significant testing grounds for 4-D regulators of $\chi$GT.