  The mechanism of superconductivity in materials with aborted ferroelectricity
and the emergence of a dilute metallic phase in systems like doped-SrTiO$_3$
are outstanding issues in condensed matter physics. This dilute metal has
features both similar and different to those found in the normal state of other
unconventional superconductors. We have investigated the optical properties of
the dilute metallic phase in doped-SrTiO$_3$ using THz time-domain
spectroscopy. At low frequencies the THz response exhibits a Drude-like form as
expected for typical metal-like conductivity. We observed the frequency and
temperature dependencies to the low energy scattering rate $\Gamma(\omega, T)
\propto (\hbar\omega)^2 + (p \pi k_BT)^2 $ expected in a conventional Fermi
liquid, despite the fact that densities are too small to allow current decay
through electron-electron scattering. However we find the lowest known $p$
values of 0.39-0.72. As $p$ is 2 in a canonical Fermi liquid and existing
models based on energy dependent elastic scattering bound $p$ from below to 1,
our observation lies outside current explanation. Our data also gives insight
into the high temperature regime and shows that the temperature dependence of
the resistivity derives in part from strong T dependent mass renormalizations.
