Vectorlike gauge theories with almost massless fermions on the lattice

A truncation of the overlap (domain wall fermions) is studied and a criterion for the reliability of the approximation is obtained by comparison to the exact overlap formula describing massless quarks. We also present a truncated version of regularized, pure gauge, supersymmetric models. The mechanism for generating almost masslessness is shown to be a generalized seesaw which can also be viewed as a version of the Froggatt-Nielsen method for obtaining natural large mass hierarchies. Viewed in this way the mechanism preserving the mass hierarchy naturally avoids preserving even approximately axial U(1). The new insights into the source of the mass hierarchy suggest ways to increase the efficiency of numerical simulations of QCD employing the truncated overlap.