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Here I$_{S}$ ( k,p) is the irreducible vertex in the particle-particle channel with total momentum zero, which means it contains all diagrams which cannot be cut into two parts by cutting two particle lines (right-ward going lines in the convention of Fig. (2)). G ( p , ω ) is the single-particle Green function at momentum-energy p , ω . In (2), and fig.(2), the integral over energy has a cut-off energy of the vertex I$_{S}$ . This cut-off ω$_{c}$ is the upper energy scale of the collective fluctuations of the fermions or of other modes with which the fermions interact. For problems of interest, we expect this cut-off energy to be much smaller than E$_{f}$ but much larger than T$_{c}$ . This can in general only be justified aposteriori . 

Quite generally, for the e-ph or the e-e problem, one can define a four-fermion inter- action vertex I$_{S}$ ( k, k + q ) which is irreducible in the particle-particle channel through the interaction Hamiltonian 

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F $_{α,β,γ,δ}$( q , ω$_{n}$ ) is the propagator of the fluctuations which are exchanged by the fermions, ω$_{n}$ sums over the boson Matsubara frequencies, and g$_{α,β}$ ( k , k + q , ν ) is the interaction vertex with ν denoting any additional idex, such as the polarization for phonons, needed to specify the coupling to the fluctuations. For interactions with phonons or any other density or current fluctuations α = β ; γ = δ . For isotropic interactions with spin-fluctuations or spin-current fluctuations, the spin-dependence in the fermion operators and the Fluctuation propagator in has the form σ$_{γ,δ}$ · σ$_{β,α}$ . 

In general three integral equations specify the Eliashberg solution: the equation for the normal self-energy of fermions, the equation for the pairing self-energy of the fermions, and the equation for the self-energy of the Bosons due to superconductivity. The latter is unimportant for the e-ph problem because the corrections due to superconductivity are of order ( m/M ) 1 / $^{2}$. But they are likely to be very important for e-e problem below T$_{c}$ , because superconductivity itself changes the spectrum of collective e-e fluctuations by gapping the low energy single-particle spectrum. This will have quite significant effects, for example on the temperature dependence of the superconducting gap. However, since we are interested here only in T$_{c}$ and that too only in a mean-field approximation, this will be neglected. 

