  The low-energy (intraband) range of the third harmonic generation of graphene
in the terahertz regime is governed by the damping terms induced by the
interactions. A controlled many-body description of the scattering processes is
thus a compelling and desirable requirement. In this paper, using a
Kadanoff-Baym approach, we systematically investigate the impact of many-body
interaction on the third-harmonic generation (THG) of graphene, taking elastic
impurity scattering as a benchmark example. We predict the onset in the mixed
inter-intraband regime of novel incoherent features driven by the interaction
at four- and five-photon transition frequencies in the third-harmonic optical
conductivity with a spectral weight proportional to the scattering rate.We show
also that, in spite of the complex many-body physics, the purely intraband term
governing the limit $\omega \to 0$ resembles the constraints of the
phenomenological model. We ascribe this agreement to the fulfilling of the
conservation laws enforced by the conserving approach. The overlap with novel
incoherent features and the impact of many-body driven multi-photon vertex
couplings limit however severely the validity of phenomenological description.

Mesoscale and Nanoscale Physics