I know that Coulomb interaction renormalizes coupling constant in BCS theory (theory with contact instantaneous interaction between 4 fermions). But I would like to see more rigorous derivation of this fact (in my problems books there is only a poor description of compensation of Coulomb repulsion). In addition, I only find really old books & papers which relate to this topic and they are unreadable.
In my view, it seems possible to start from the action for BCS theory and just add Coulomb term, $$S_{\text{int}}=\int d\tau\int d^3x\left[-g\bar{\psi}\bar{\psi}\psi\psi+\bar{\psi}\psi V(x)\bar{\psi}\psi\right],$$ where I omit spin indices of fermion fields and $V(x)$ is Coulomb potential. So, it seems that to decouple this ineraction I need two channels. But I am not sure that decoupling in two channels simultaneously is neccessary and the problem with two channels seems hard to solve.
Moreover, I am not sure that bare Coulomb is right interaction. In my view, screened Coloumb interaction is more appropriate. If it is true, in case of low transfered momentum $q$ in screened Coulomb $V(q)$, I can neglect momentum in denominator and find that interaction becomes contact. Okey, may be it is useful but with these facts I have $$-g\bar{\psi}\bar{\psi}\psi\psi+g_{\text{Cl}}\bar{\psi}\psi\bar{\psi}\psi$$ where $g_{\text{Cl}}$ means screened Coulomb in the limit of low transfered momentum and this expression seems right (roughly speaking, effective attraction constant is $-g+g_{\text{Cl}}$) but spin structure of this two terms are different and I again return to two channels decoupling. Finally, my approximations seems really rough.
So, can anybody give some refs & comments on compensation of Coulomb repulsion in BCS superconductivity theory in modern (field theoretical) description?
My attempd was decoupling in two channels simultaneously. I have considered fermions with Coulomb interaction and introduce Hubbard-Stratanovich field $\phi$. This fild modifies kinetic term of fermions and also has kinetic term $(\partial\phi)^2/8\pi$. Then, due to modification of kinetic term, I can easily decouple in Cooper channel, too. Superconductor Green function obtains shift in temporal component by $ie\phi(\tau,x)$ which coincides with external field. Finally, I obtain following effective action, $$\frac{(\partial\phi)^2}{8\pi}+\frac{|\Delta|^2}{g}+\frac{1}{g}\mathrm{tr}\ln(1+\mathcal{G}_{0}\Delta),$$ where $$\mathcal{G}_0=\begin{pmatrix}-\partial_{\tau}-ie\phi(\tau,x)+\frac{\partial^2}{2m}+\mu & 0 \\ 0 & -\partial_{\tau}+ie\phi(\tau,x)-\frac{\partial^2}{2m}-\mu\end{pmatrix}.$$ But I do not see what should I do to see that effective field $\phi$ renormalizes attractive coupling $g$.
I believe that this question will useful for many other people.
