A weakly correlated many-electron system can be viewed in a first approximation as a Fermi liquid, meaning that it behaves similarly to a non-interacting electron gas with renormalized parameters.
In this respect, one can calculate the electronic self-energy $\Sigma_{\text{el}}(k, \omega)$ which real and imaginary parts give information about the low-lying excitations of the system (the so-called quasiparticles).
Given an arbitrary interaction between electrons, do we have a general criterion on what the electronic self-energy should be, or how it should behave, for the system to be accurately described by the Fermi liquid theory ? I've heard a lot of awnsers to this question such has "the self-energy must be a smooth varying function" or "one must be able to expand it in powers of $k$ and $\omega$" but I am somewhat not satisfied with these awnsers, and was wondering if there existed a more precise criterion.
Any concrete example with an existing model would be greatly appreciated.