  The celebrated variational path integral approach to the polaron problem
shows remarkable discrepancies with diagrammatic Monte Carlo for the
Bogoliubov-Fr\"{o}hlich Hamiltonian which describes an impurity weakly coupled
to a Bose condensed atomic gas. It has been shown both via a renormalization
group approach and by the method of correlated Gaussian wavefunctions that the
model has a subtle UV divergence caused by quantum fluctuations, which are not
captured within Feynman's approach. In this work we address the issues with
Feynman's approach and show that by extending the model action to a more
general form, and by considering higher order corrections beyond the
Jensen-Feynman inequality, a good agreement with diagrammatic Monte Carlo can
be obtained.
