  Motivated by the mid-infrared scenario for high-temperature superconductivity
proposed by Leggett, the effects of Umklapp processes on the density-density
correlation function in the presence of long-range Coulomb interaction have
been investigated on a microscopic model. We show that because Umklapp
processes enable scatterings that conserve total momentum only up to
$n\vec{K}$, where $n$ is an integer and $\vec{K}$ is the reciprocal wave
vector, significant amounts of spectral weight in the plasmonic excitations at
long wavelength are transferred into lower frequency around mid-infrared
regime. We further find that regardless of the gap symmetry, superconductivity
generally suppresses the Umklapp scatterings due to the formation of the
electron pairing. This suppression is unique for the superconductivity due to
the interplay between electron pairing and the odd parity of the matrix
elements associated with Umklapp channels, which usually does not occur in
other known competing orders. Specific predictions for the experimental
signatures in optical conductivity and electron energy loss spectroscopy (EELS)
will be discussed.
