  Recent scanning tunneling spectroscopy experiments [V. W. Brar et. al., Appl.
Phys. Lett. 91, 122102 (2007), Y. Zhang et. al., arXiv:0802.4315 (2008)] on
graphene reported an unexpected gap of about $\pm 60$meV around the Fermi
level. Here, we give a theoretical investigation explaining the experimentally
observed spectra and confirming the phonon mediated tunneling as the reason for
the gap: We study the real space properties of the wave functions involved in
the tunneling process by means of ab-initio theory and present a model for the
electron-phonon interaction, which couples the graphene's Dirac electrons with
quasi free electron states at the Brillouin zone center. The self-energy
associated with this electron-phonon interaction is calculated and its effects
on tunneling into graphene are discussed. In particular, good agreement of the
tunneling density of states within our model and the experimental[V. W. Brar
et. al., Appl. Phys. Lett. 91, 122102 (2007), Y. Zhang et. al., arXiv:0802.4315
(2008)] dI/dU spectra is found.
