  Motivated by experiments with cold atoms, we investigate a mobile impurity
immersed in a Fermi sea in three dimensions at zero temperature by means of the
functional renormalization group. We first perform the derivative expansion of
the effective action to calculate the ground state energy and Tan's contact
across the polaron-molecule transition for several mass imbalances. Next we
study quasiparticle properties of the impurity by using a real-time method
recently developed in nuclear physics, which allows one to go beyond the
derivative expansion. We obtain the spectral function of the polaron, the
effective mass and quasiparticle weight of attractive and repulsive polarons,
and clarify how they are affected by mass imbalances.
