  The Mott metal-insulator transition is a typical strong correlation effect
triggered by the Umklapp scattering. However, in a physical system, the Umklapp
scattering coexists with the normal scattering, including both forward and
backward scattering, which conserves the total momentum of scattered electrons.
Therefore, it is not easy to quantify the contribution of the Umklapp
scattering in a Mott metal-insulator transition. To resolve this difficulty, we
propose to explore these scattering processes separately. We study the
contribution of each scattering process in the one-dimensional Hubbard model
using the momentum-space density-matrix renormalization group (kDMRG) and
bosonization methods. Our kDMRG calculation confirms that the Mott charge gap
results from the Umklapp scattering, but the normal scattering processes
strongly renormalize its value. Furthermore, we present a scaling analysis of
the Mott charge gap in the bosonization theory and show that the interplay
between the Umklapp and forward scattering dictates the charge dynamics in the
half-filled Hubbard model.
