  We theoretically examine the finite-temperature spectral function of Fermi
polarons in three dimensions, by using a self-consistent many-body $T$-matrix
theory in real frequency. In comparison with the previous results from a
non-self-consistent many-body $T$-matrix approach, we show that the treatment
of self-consistency in the impurity Green function leads to notable changes in
almost all the dynamical quantities, including the vertex function, impurity
self-energy and spectral function. Eventually, it gives rise to quantitatively
different predictions for the measurable radio-frequency spectrum and Raman
spectrum at finite temperature. Using the recent spectroscopic measurements as
a benchmark, we find that the self-consistent many-body $T$-matrix theory
somehow provides a better explanation for the experimental data. The notable
difference in the predictions from the non-self-consistent and self-consistent
theories suggests that more accurate theoretical descriptions are needed, in
order to fully account for the current spectroscopic observations on Fermi
polarons.
