The electron many-body problem in graphene

We give a brief summary of the current status of the electron many-body problem in graphene. We claim that graphene has intrinsic dielectric properties which should dress the interactions among the quasiparticles, and may explain why the observation of electron-electron renormalization effects has been so elusive in the recent experiments. We argue that the strength of Coulomb interactions in graphene may be characterized by an effective fine structure constant given by $\alpha^{\star}(\mathbf{k},\omega)\equiv2.2/\epsilon(\mathbf{k},\omega)$, where $\epsilon(\mathbf{k},\omega)$ is the dynamical dielectric function. At long wavelengths, $\alpha^{\star}(\mathbf{k},\omega)$ appears to have its smallest value in the static regime, where $\alpha^{\star}(\mathbf{k}\to0,0)\approx1/7$ according to recent inelastic x-ray measurements, and the largest value in the optical limit, where $\alpha^{\star}(0,\omega)\approx2.6$. We conclude that the strength of Coulomb interactions in graphene is not universal, but depends highly on the scale of the phenomenon of interest. We propose a prescription in order to reconcile different experiments.