  We analyze the properties of an impurity in a dilute Bose-Einstein condensate
(BEC). First the quasiparticle residue of a static impurity in an ideal BEC is
shown to vanish with increasing particle number as a stretched exponential,
leading to a bosonic orthogonality catastrophe. Then we introduce a variational
ansatz, which recovers this exact result and describes the macroscopic dressing
of the impurity including its back-action onto the BEC as well as boson-boson
repulsion beyond the Bogoliubov approximation. This ansatz predicts that the
orthogonality catastrophe also occurs for mobile impurities, whenever the BEC
becomes ideal. Finally, we show that our ansatz agrees well with experimental
results.
