Mean-field construction for spectrum of one-dimensional Bose polaron

The full momentum dependence of spectrum of a point-like impurity immersed in a dilute one-dimensional Bose gas is calculated on the mean-field level. In particular we elaborate, to the finite-momentum Bose polaron, the path-integral approach whose semi-classical approximation leads to the conventional mean-field treatment of the problem while quantum corrections can be easily accounted by standard loop expansion techniques. The extracted low-energy parameters of impurity spectrum, namely, the binding energy and the effective mass of particle, are shown to be in qualitative agreement with the results of quantum Monte Carlo simulations.