  We present {\it ab-initio} time-dependent density-functional theory
calculation results for low-energy collective electron excitations in
$\textrm{MgB}_2$. The existence of a long-lived collective excitation
corresponding to coherent charge density fluctuations between the boron
$\sigma$- and $\pi$- bands ($\sigma\pi$ mode) is demonstrated. This mode has a
sine-like oscillating dispersion for energies below 0.5 eV. At even lower
energy we find another collective mode ($\sigma\sigma$ mode). We show the
strong impact of local-field effects on dielectric functions in MgB$_2$. These
effects account for the long q-range behavior of the modes. We discuss the
physics that these collective excitations bring to the energy region typical
for lattice vibrations.
