I am trying to understand the computational methods to calculate the spectral line-width as done in this paper, http://www.nature.com/articles/ncomms11755
Here, they say that the line-width is calculated as the imaginary part of the electron-phonon self energy $(2 Im(\Sigma))$. However, I don't understand how you can get $(2 Im(\Sigma))$ for the line-width using the Fermi's golden rule.
Does anyone know how to show that the line-width is given by $(2 Im(\Sigma))$? Or a good reference which shows hot to do this?