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The anomalous features of the Raman spectroscopy measurement in MgB$_2$represent a still unresolved puzzle. In particular highly debated are theorigin of the huge $E_{2g}$ phonon linewidth, the nature of the low energy($\omega < \omega_{E_{2g}}$) background and the evolution of the Raman spectrawith Al doping. In this paper we compute the self-energy of the $E_{2g}$ phononmode in a fully self-consistent way taking into account electron-phonon effectson the electronic properties. We show that all the anomalous features can benaturally understood in a framework where the whole electron-phonon spectrum$\alpha^2F(\omega)$ gives rise to significant damping processes for theelectronic excitations and consequently for the $E_{2g}$ phonon itself. Thetwo-peak structure as function of the Al doping is ascribed to finite bandwidtheffects arising as the Fermi level approaches the $\sigma$ band edge.Comment: 4 revtex pages, eps figures included, final version as appears in  Phys. Rev. B (Rapid Communication

physical review. b, condensed matter and materials physics

Electron-phonon effects on the Raman spectrum in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi mathvariant="normal">Mg</mml:mi><mml:msub><mml:mi mathvariant="normal">B</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math>

Electron-phonon effects on the Raman spectrum inMgB2