  The damping rate of two-dimensional massless Dirac fermions exhibit non-Fermi
liquid behavior, $\propto \epsilon^{1/2}$, due to gauge field at zero
temperature and zero chemical potential. We study the fate of this behavior at
finite chemical potential. We fist calculate explicitly the temporal and
spatial components of vacuum polarization functions. The analytical expressions
imply that the temporal component of gauge field develops a static screening
length at finite chemical potential while the transverse component remains
long-ranged owing to gauge invariance. We then calculate the fermion damping
rate and show that the temporal gauge field leads to normal Fermi liquid
behavior but the transverse gauge field leads to non-Fermi liquid behavior
$\propto \epsilon^{2/3}$ at zero temperature. This energy-dependence is more
regular than $\propto \epsilon^{1/2}$ and does not change as chemical potential
varies.
