  Background: The electric dipole strength detected around the particle
threshold and commonly associated to the pygmy dipole resonance offers a unique
information on neutron skin and symmetry energy, and is of astrophysical
interest. The nature of such a resonance is still under debate.
  Purpose: We intend to describe the giant and pygmy resonances in 208 Pb by
enhancing their fragmentation with respect to the random-phase approximation.
  Method: We adopt the equation of motion phonon method to perform a fully
self-consistent calculation in a space spanned by one-phonon and two-phonon
basis states using an optimized chiral two-body potential. A phenomenological
density dependent term, derived from a contact three-body force, is added in
order to get single-particle spectra more realistic than the ones obtained by
using the chiral potential only. The calculation takes into full account the
Pauli principle and is free of spurious center of mass admixtures.
  Results: We obtain a fair description of the giant resonance and obtain a
dense low-lying spectrum in qualitative agreement with the experimental data.
The transition densities as well as the phonon and particle-hole composi- tion
of the most strongly excited states support the pygmy nature of the low-lying
resonance. Finally, we obtain realistic values for the dipole polarizability
and the neutron skin radius.
  Conclusions: The results emphasize the role of the two-phonon states in
enhancing the fragmentation of the strength in the giant resonance region and
at low energy, consistently with experiments. For a more detailed agreement
with the data, the calculation suggests the inclusion of the three-phonon
states as well as a fine tuning of the single-particle spectrum to be obtained
by a refinement of the nuclear potential.
