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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?

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In QFT the scattering amplitude $S=\mathbb 1 + i T$ is unitary and hence fulfills

$$\mathbb 1 =(\mathbb 1+iT)^\dagger (\mathbb 1+iT)= 1 + iT -iT^\dagger +T^\dagger T .$$

This gives $\sum_B |T_{AB}|^2 = 2 \operatorname{Im}(T_{AA})$ and is known as the optical theorem.

One of the most known applications is to compute the scattering amplitude ($e^+ e^- \rightarrow \bar{q} q$) in QCD by considering the loop diagrams ($e^+ e^- \rightarrow e^+ e^-$) with intermediate hadron loops in the middle.

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For the wanted electrons to hadrons amplitude many possible configurations have to be considered. The quark loop is indeed not just one possible quark loop but a quark gluon self energy $\Pi_h$ such that the cross section would read $\sigma =const \times\frac{1}{s} \operatorname{Im}(\Pi_h)$.

I am not quite sure about the exact application of this in condensed matter physics but I can imagine that is what is being used in the article.

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