Tunneling into quantum wires: regularization of the tunneling Hamiltonian and consistency between free and bosonized fermions

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.