content_image/415062.jpg content_image/415063.jpg content_image/415064.jpg content_image/415065.jpg content_image/415066.jpg content_image/415067.jpg content_image/415068.jpg content_image/415069.jpg content_image/415070.jpg
where X ∈ { G, G $^{†}$, F, F $^{†}$} and g is the degeneracy factor, which we omit in the intermediate equations and restore in the final ones. Direct calculations show that 

<img src='content_image/415062.jpg'>

The remaining equations read 

these equalities imply that Γ$_{1}$ = Γ$_{4}$, which is a conse- quence of the time-reversal invariance of the system. 

<img src='content_image/415063.jpg'>

There are six distinct loops in Eq. (11), namely Π$_{GG}$, Π$_{FF}$ , Π$_{GF}$ , Π$_{FG}$, Π$_{G}$ † $_{G}$, Π$_{GG}$ † . In the weak coupling limit Π$_{GF}$ ≃ − Π$_{FG}$, Π$_{G}$ $^{†}$G ≃ Π$_{GG}$ † , an approximation discussed in detail in Sec. II B. This reduces the number of equations from three to two 

where 

with q = ( ω, q ). The solutions for the remaining two vertex functions reads 

<img src='content_image/415064.jpg'>

<img src='content_image/415065.jpg'>

<img src='content_image/415066.jpg'>

<img src='content_image/415067.jpg'>

<img src='content_image/415068.jpg'>

<img src='content_image/415069.jpg'>

When v$_{pp}$ = v$_{ph}$ these reduce to Eqs. (16) and (17) of Ref. [2]. It seen that the interaction in the pp channel is absorbed in the gap equation and it is the ph interaction that enters the renormalization of the one-loop polariza- tion tensor. This result could have been anticipated from the limiting form of the RPA polarization tensor of nor- mal Fermi liquids (see Sec. III C). 

The full polarization tensor is given by Eq. (35) of Ref. [2] with the replacement v → v$_{ph}$ 

<img src='content_image/415070.jpg'>

## B. Polarization tensor 

