3 these equalities imply that Γ$_{1}$ = Γ$_{4}$, which is a conse- quence of the time-reversal invariance of the system. 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 The remaining equations read Π$_{G}$ $^{†}$F = Π$_{FG}$ , Π$_{GF}$ = Π$_{FG}$ † , Π$_{G}$ $^{†}$G† = Π$_{GG}$; (10)   1 − v$_{ph}$ [Π$_{GG}$ − Π$_{FF}$ ] v$_{ph}$ Π$_{GF}$ v$_{ph}$ Π$_{FG}$ − 2 v$_{pp}$ Π$_{GF}$ [1 − v$_{pp}$ Π$_{GG}$ † ] v$_{pp}$ Π$_{FF}$ 2 v$_{pp}$ Π$_{FG}$ v$_{pp}$ Π$_{FF}$ [1 − v$_{pp}$ Π$_{G}$ † $_{G}$]     Γ$_{1}$ Γ$_{2}$ Γ$_{3}$   =   Γ$_{0}$ 0 0   . (11) ( 1 − v$_{ph}$ A v$_{ph}$ B v$_{pp}$ B − v$_{pp}$ C ) ( Γ$_{1}$ Γ$_{2}$ ) = ( Γ$_{0}$ 0 ) , (12) A ( q ) = Π$_{GG}$( q ) − Π$_{FF}$ ( q ) , (13) B ( q ) = 2Π$_{FG}$( q ) , (14) C ( q ) = Π$_{GG}$ † ( q ) + Π$_{FF}$ ( q ) − ( v$_{pp}$ ) − $^{1}$, (15) Γ$_{1}$( q ) = Γ$_{0}$ C ( q ) C ( q ) − v$_{ph}$ [ A ( q ) C ( q ) + B ( q ) $^{2}$] , (16) Γ$_{2}$( q ) = − Γ$_{0}$ B ( q ) C ( q ) − v$_{ph}$ [ A ( q ) C ( q ) + B ( q ) $^{2}$] . (17) Π $^{R}$( q ) = A ( q ) C ( q ) + B ( q ) 2 C ( q ) − v$_{ph}$ [ A ( q ) C ( q ) + B ( q ) $^{2}$] . (18) B. Polarization tensor 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 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). where with q = ( ω, q ). The solutions for the remaining two vertex functions reads The full polarization tensor is given by Eq. (35) of Ref. [2] with the replacement v → v$_{ph}$