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We develop a theory for the electron-phonon interaction effects on theelectronic properties of graphene. We analytically calculate the electronself-energy, spectral function and band velocity renormalization due tophonon-mediated electron-electron interaction. We find that phonon-mediatedelectron-electron coupling has a large effect on the graphene band structurerenormalization, and our analytic theory successfully captures the essentialfeatures of the observed graphene electron spectra in the ARPES experiments,predicting a kink at $\sim 200\mathrm{meV}$ below the Fermi level and areduction of the band velocity by $\sim 10-20%$ at the experimental dopinglevel.

Phonon-induced many-body renormalization of graphene electronic properties