We study excitonic states of an atomic impurity in a Fermi gas, i.e., bound states consisting of the impurity and a hole. Previous studies considered bound states of the impurity with particles from the Fermi sea where the holes only formed part of the particle-hole dressing. Within a two-channel model, we find that, for a wide range of parameters, excitonic states are not ground but metastable states. We further calculate the decay rates of the excitonic states to polaronic and dimeronic states and find they are long lived, scaling as $Gamma^{rm{Exc}}_ {rm{Pol}} propto ( Deltaomega)^{5.5}$ and $Gamma^{rm{Exc}}_ {rm{Dim}} propto (Deltaomega)^{4}$. We also find that a new continuum of exciton-particle states should be considered alongside the previously known dimeron-hole continuum in spectroscopic measurements. Excitons must therefore be considered as a new ingredient in the study of metastable physics currently being explored experimentally.
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