Electron-phonon effects on the Raman spectrum in MgB$_2$
  The anomalous features of the Raman spectroscopy measurement in MgB$_2$ represent a still unresolved puzzle. In particular highly debated are the origin of the huge $E_{2g}$ phonon linewidth, the nature of the low energy ($\omega < \omega_{E_{2g}}$) background and the evolution of the Raman spectra with Al doping. In this paper we compute the self-energy of the $E_{2g}$ phonon mode in a fully self-consistent way taking into account electron-phonon effects on the electronic properties. We show that all the anomalous features can be naturally understood in a framework where the whole electron-phonon spectrum $\alpha^2F(\omega)$ gives rise to significant damping processes for the electronic excitations and consequently for the $E_{2g}$ phonon itself. The two-peak structure as function of the Al doping is ascribed to finite bandwidth effects arising as the Fermi level approaches the $\sigma$ band edge. 
E. Cappelluti