  We present a systematic study of a mobile impurity immersed in a
three-dimensional Fermi sea of fermions at finite temperature, by using the
standard non-self-consistent many-body $T$-matrix theory that is equivalent to
a finite-temperature variational approach with the inclusion of
one-particle-hole excitation. The impurity spectral function is determined in
the real-frequency domain, avoiding any potential errors due to the numerical
analytic continuation in previous $T$-matrix calculations and the small
spectral broadening parameter used in variational calculations. In the
weak-coupling limit, we find that the quasiparticle decay rate of both
attractive and repulsive polarons does not increase significantly with
increasing temperature, and therefore Fermi polarons may remain well-defined
far above Fermi degeneracy. In contrast, near the unitary limit with strong
coupling, the decay rate of Fermi polarons rapidly increase and the
quasiparticle picture breaks down close to the Fermi temperature. We analyze in
detail the recent ejection and injection radio-frequency (rf) spectroscopy
measurements, performed at Massachusetts Institute of Technology (MIT) and at
European Laboratory for Non-Linear Spectroscopy (LENS), respectively. We show
that the momentum average of the spectral function, which is necessary to
account for the observed rf-spectroscopy, has a sizable contribution to the
width of the quasiparticle peak in spectroscopy. As a result, the measured
decay rate of Fermi polarons could be significantly larger than the calculated
quasiparticle decay rate at zero momentum. By take this crucial contribution
into account, we find that there is a reasonable agreement between theory and
experiment for the lifetime of Fermi polarons in the strong-coupling regime, as
long as they remain well-defined.
