# How to prove that the spectral line-width is given by the imaginary part the self energy?

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?

• Hint: line frequency width would be zero if all $C$'s in factors $e^{i C t/\hbar}$ in expression for probability of original state were real numbers. $C$ has to have some imaginary component for the probability to decay in time. Since $C$ has dimensions of energy, it can be interpreted (does not need to be) as part of "self-energy", especially in some methods of calculation which use the concept. – Ján Lalinský Jul 28 '19 at 18:07

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. 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.