Effect of Holstein phonons on the electronic properties of graphene

We obtain the self-energy of the electronic propagator due to the presence of Holstein polarons within the first Born approximation. This leads to a renormalization of the Fermi velocity of 1%. We further compute the optical conductivity of the system at the Dirac point and at finite doping within the Kubo formula. We argue that the effects due to Holstein phonons are negligible and that the Boltzmann approach, which does not include inter-band transitions and can thus not treat optical phonons due to their high energy of (h) over barw(0) similar to 0.1-0.2 eV, remains valid.