  We discuss the dynamical polarization with finite momentum and frequency in
the presence of many-electron effect, including the screened Coulomb
interaction, self-energy and vertex correction. The longitudinal conductivity,
screened Coulomb interaction, and the response function are calculated. The
behavior of the Dirac Fermions, including the propagation of the charge density
which exhibits the causality, affects largely the low-temperature physical
properties of the Dirac semimetal, like the silicene. For the
polarization-related quantities (like the dielectric function), the method of
standard random phase approximation (RPA) provides the non-interaction results
(ignore the many electron effect), for a more exact result, we discuss the
self-energy and the vertex correction for the two-dimension Dirac model. We
found that, after the self-energy correction, the longitudinal conductivity
increase compared to the noninteracting one in optical limit. For the
renormalization treatment, the ultraviolet cutoff is setted as $\Lambda=t$ in
our calculations, i.e., within the range between two Van Hove singularities
where the density of states divergent logarithmically. %The $\Lambda$ is indeed
a non-universal parameter and can also be choiced larger than arbitary ${\bf
q}$ or $\omega$. The (corrected) screened Coulomb interaction and the response
function are also discussed. Our results are helpful to the application of the
Dirac materials (or the Weyl semimetal) in spintronics or valleytronics.
