  We study the supersymmetric kink with higher derivative and momentum cut-off
regularization schemes. We introduce the new momentum cut-off regularization
scheme which we call ``generalized momentum cut-off''. A new, explicit
computation for the central charge anomaly for this scheme is described in
detail. The calculation of the first order mass corrections for the bosonic and
supersymmetric kink within the momentum cut-off is presented in hep-th/0010051,
so that one can compare the one-loop central charge and mass computed
independently within the same regularization setup. We confirm that the BPS
bound is saturated in one loop level. Thus the Wilsonian momentum cut-off
regularization scheme is rehabilitated as a bona fide procedure for computing
quantum corrections in the topologically nontrivial backgrounds. We lead the
reader to the idea that a consistent regularization in general does not only
assume that one regulates loops properly, but also requires caution in defining
the total number of modes involved in the quantization of the theory. We also
study the higher-derivative regularization scheme of Shifman et al. in great
detail. We extend the Noether method for the case when the higher-derivatives
are present in the Lagrangian. The extensive discussion of the technical
aspects and the consistency of the higher-derivative scheme is given. We show
that higher-derivative regularization gives, in general, a nonlocal topological
current, which leads to the correct value of the anomaly in the central charge.
We also comment on the status of the dimensional regularization approach to the
computation of the central charge anomaly and kink mass.
