  We study the properties of an impurity of mass $M$ moving through a spatially
homogeneous three-dimensional fully polarized Fermi gas of particles of mass
$m$. In the weakly attractive limit, where the effective coupling constant
$g\to0^-$ and perturbation theory can be used, both for a broad and a narrow
Feshbach resonance, we obtain an explicit analytical expression for the complex
energy $\Delta E(\KK)$ of the moving impurity up to order two included in $g$.
This also gives access to its longitudinal and transverse effective masses
$m_\parallel^*(\KK)$, $m_\perp^*(\KK)$, as functions of the impurity wave
vector $\KK$. Depending on the modulus of $\KK$ and on the impurity-to-fermion
mass ratio $M/m$ we identify four regions separated by singularities in
derivatives with respect to $\KK$ of the second-order term of $\Delta E(\KK)$,
and we discuss the physical origin of these regions. Remarkably, the
second-order term of $m_\parallel^*(\KK)$ presents points of
non-differentiability, replaced by a logarithmic divergence for $M=m$, when
$\KK$ is on the Fermi surface of the fermions. We also discuss the third-order
contribution and relevance for cold atom experiments.
