  An impurity immersed in a Bose-Einstein condensate is no longer accurately
described by the Fr\"ohlich Hamiltonian as the coupling between the impurity
and the boson bath gets stronger. We study the dominant effects of the
two-phonon terms beyond the Fr\"ohlich model on the ground-state properties of
the polaron using Feynman's variational path-integral approach. The previously
reported discrepancy in the effective mass between the renormalization group
approach and this theory is shown to be absent in the beyond-Fr\"ohlich model
on the positive side of the Feshbach resonance. Self-trapping, characterized by
a sharp and dramatic increase of the effective mass, is no longer observed for
the repulsive polaron once the two-phonon interactions are included. For the
attractive polaron we find a divergence of the ground-state energy and
effective mass at weaker couplings than previously observed within the
Fr\"ohlich model.
