Diagrammatic weak-coupling expansion for the magneto-polaron energy

The problem of a Pekar-Fröhlich polaron in a magnetic field is considered for weak electron-phonon interaction. A diagrammatic technique for zero temperature is developed. Previously obtained results for two dimensions are reproduced. For three dimensions the polaron ground-state energy and the longitudinal effective mass are derived in the first orders of the electron-phonon coupling constant for non-zero values of a magnetic field. In a strong magnetic field the bulk polaron is shown to be equivalent to a one-dimensional polaron with a renormalized electron-phonon coupling constant. This leads to exact results in the strong-coupling limit.