Electron-phonon renormalization in small Fermi energy systems

The puzzling features of recent photoemission data in cuprates have been object of several analysis in order to identity the nature of the underlying electron-boson interaction. In this paper we point out that many basilar assumptions of the conventional analysis as expected to fail in small Fermi energy systems when, as the cuprates, the Fermi energy $E_{\mathrm{F}}$ is comparable with the boson energy scale. We discuss in details the novel features appearing in the self-energy of small Fermi energy systems and the possible implications on the ARPES data in cuprates.