  Tunneling between a point contact and a one-dimensional wire is usually
described with the help of a tunneling Hamiltonian that contains a delta
function in position space. Whereas the leading order contribution to the
tunneling current is independent of the way this delta function is regularized,
higher-order corrections with respect to the tunneling amplitude are known to
depend on the regularization. Instead of regularizing the delta function in the
tunneling Hamiltonian, one may also obtain a finite tunneling current by
invoking the ultraviolet cut-offs in a field-theoretic description of the
electrons in the one-dimensional conductor, a procedure that is often used in
the literature. For the latter case, we show that standard ultraviolet cut-offs
lead to different results for the tunneling current in fermionic and bosonized
formulations of the theory, when going beyond leading order in the tunneling
amplitude. We show how to recover the standard fermionic result using the
formalism of functional bosonization and revisit the tunneling current to
leading order in the interacting case.
