  We study theoretically the lifetimes of attractive and repulsive Fermi
polarons, as well as the molecule at finite momentum in three dimensions. To
this end, we develop a new technique that allows for the computation of Green's
functions in the whole complex frequency plane using exact analytical
continuation within the functional renormalization group. The improved
numerical stability and reduced computational cost of this method yield access
to previously inaccessible momentum-dependent quasiparticle properties of
low-lying excited states. While conventional approaches like the
non-selfconsistent $T$-matrix approximation method cannot determine these
lifetimes, we are able to find the momentum-dependent lifetime at different
interaction strengths of both the attractive and repulsive polaron as well as
the molecule. At weak coupling our results confirm predictions made from
effective Fermi liquid theory regarding the decay of the attractive polaron,
and we demonstrate that Fermi liquid-like behavior extends far into the
strong-coupling regime where attractive polaron and molecule exhibit a $p^4$
momentum scaling in their decay widths. Our results offer an intriguing insight
into the momentum-dependent quasiparticle properties of the Fermi polaron
problem, which can be measured using techniques such as Raman transfer and
Ramsey interferometry.
