  Background: Models based on using perturbative polarization corrections and
mean-field blocking approximation give conflicting results for masses of odd
nuclei.
  Purpose: Systematically investigate the polarization and mean-field models,
implemented within self-consistent approaches that use identical interactions
and model spaces, so as to find reasons for the conflicts between them.
  Methods: For density-dependent interactions and with pairing correlations
included, we derive and study links between the mean-field and polarization
results obtained for energies of odd nuclei. We also identify and discuss
differences between the polarization-correction and full
particle-vibration-coupling (PVC) models. Numerical calculations are performed
for the mean-field ground-state properties of deformed odd nuclei and then
compared to the polarization corrections determined by using the approach that
conserves spherical symmetry.
  Results: We have identified and numerically evaluated self-interaction (SI)
energies that are at the origin of different results obtained within the
mean-field and polarization-correction approaches.
  Conclusions: Mean-field energies of odd nuclei are polluted by the SI
energies, and this makes them different from those obtained by using
polarization-correction methods. A comparison of both approaches allows for the
identification and determination of the SI terms, which then can be calculated
and removed from the mean-field results, giving the self-interaction-free
energies. The simplest deformed mean-field approach that does not break parity
symmetry is unable to reproduce full PVC effects.
