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I have posted several questions about superconductivity recently and all of them are related to vertex function but these questions were incorrect.

I have found the following statement in book written by Abrikosov A.A., Gorkov L.P., Dzyaloshinskii I.E.. If one consider BCS-hamiltonian with contact interaction $g$, we have $$\langle T\Psi_1\Psi_2\Psi^{\dagger}_3\Psi^{\dagger}_4\rangle$$ and this two-particle functions can be decomposed as follows $$-\langle T\Psi_1\Psi_3^{\dagger}\rangle\langle T\Psi_2\Psi_4^{\dagger}\rangle+\langle T\Psi_1\Psi^{\dagger}_4\rangle\langle T\Psi_2\Psi_3^{\dagger}\rangle+\langle N|T\Psi_1\Psi_2|N+2\rangle\langle N+2|\Psi_3^{\dagger}\Psi_{4}^{\dagger}|N\rangle.$$ This decompositions means that we neglect scattering procceses wich are contained in vertex function and we take into account only vacuum reconstruction of theory.

My question: starting from BCS-theory with coupling $g$ and then consider vertex function, can one obtain vacuum reconstruction? I know that we can see that vertex function contains pole which is related to SC-transition but is it possible to obtain more information than transition point? Honestly, I am not sure that machinery of peturbation theory by $g$ will be applicable for this goal.

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