4 The “elementary loops” are defined explicitly as Π$_{GG}$( q ) = ∫ d $^{3}$p (2 π ) 3 {[ u 2 $_{p}$u 2 k iq + ϵ$_{p}$ − ϵ$_{k}$ − v 2 $_{p}$v 2 k iq − ϵ$_{p}$ + ϵ$_{k}$ ] [ f ( ϵ$_{p}$ ) − f ( ϵ$_{k}$ )] + [ u 2 $_{k}$v 2 p iq − ϵ$_{p}$ − ϵ$_{k}$ − u 2 $_{p}$v 2 k iq + ϵ$_{p}$ + ϵ$_{k}$ ] [ f ( − ϵ$_{p}$ ) − f ( ϵ$_{k}$ )] } , (19) Π$_{FG}$( q ) = − ∫ d $^{3}$p (2 π ) 3 { u$_{p}$v$_{p}$ [ u 2 k iq + ϵ$_{p}$ − ϵ$_{k}$ + v 2 k iq − ϵ$_{p}$ + ϵ$_{k}$ ] [ f ( ϵ$_{p}$ ) − f ( ϵ$_{k}$ )] − u$_{p}$v$_{p}$ [ u 2 k iq − ϵ$_{p}$ − ϵ$_{k}$ + v 2 k iq + ϵ$_{p}$ + ϵ$_{k}$ ] [ f ( − ϵ$_{p}$ ) − f ( ϵ$_{k}$ )] } , (20) Π$_{FF}$ ( q ) = ∫ d $^{3}$p (2 π ) 3 { u$_{p}$u$_{k}$v$_{p}$v$_{k}$ { [ 1 iq + ϵ$_{p}$ − ϵ$_{k}$ − 1 iq − ϵ$_{p}$ + ϵ$_{k}$ ] [ f ( ϵ$_{p}$ ) − f ( ϵ$_{k}$ )] + [ 1 iq + ϵ$_{p}$ + ϵ$_{k}$ − 1 iq − ϵ$_{p}$ − ϵ$_{k}$ ] [ f ( − ϵ$_{p}$ ) − f ( ϵ$_{k}$ )] }} , (21) Π$_{G}$ † $_{G}$( q ) = − ∫ d $^{3}$p (2 π ) 3 {[ u 2 $_{k}$v 2 p iq + ϵ$_{p}$ − ϵ$_{k}$ − u 2 $_{p}$v 2 k iq − ϵ$_{p}$ + ϵ$_{k}$ ] [ f ( ϵ$_{p}$ ) − f ( ϵ$_{k}$ )] + [ u 2 $_{p}$u 2 k iq − ϵ$_{p}$ − ϵ$_{k}$ − v 2 $_{p}$v 2 k iq + ϵ$_{p}$ + ϵ$_{k}$ ] [ f ( − ϵ$_{p}$ ) − f ( ϵ$_{k}$ )] } , (22) I = ∫ d $^{3}$p (2 π ) 3 Δ 2 2 ϵ$_{p}$ ξ$_{p}$ ϵ$_{p}$ f ( ϵ$_{p}$ ) − f ( ϵ$_{k}$ ) iq + ϵ$_{p}$ − ϵ$_{k}$ ≃ 0 . (23) Π$_{FG}$( q ) = − 1 2 ∫ d $^{3}$p (2 π ) 3 { u$_{p}$v$_{p}$ [ 1 iq + ϵ$_{p}$ − ϵ$_{k}$ + 1 iq − ϵ$_{p}$ + ϵ$_{k}$ ] [ f ( ϵ$_{p}$ ) − f ( ϵ$_{k}$ )] − u$_{p}$v$_{p}$ [ 1 iq − ϵ$_{p}$ − ϵ$_{k}$ + 1 iq + ϵ$_{p}$ + ϵ$_{k}$ ] [ f ( − ϵ$_{p}$ ) − f ( ϵ$_{k}$ )] } . (24) where k = p + q and the coherence factors are given by u 2 p = (1 / 2)[1 + ξ$_{p}$/ϵ$_{p}$ ] and v 2 p = 1 − u 2 $_{p}$. The form of the polarization loop Π$_{G}$ † $_{G}$( q ) is valid for arbitrary couplings. The relation Π$_{G}$ † $_{G}$( q ) = Π$_{GG}$ † ( q ) is established on noting that, for example, u 2 $_{p}$v 2 k = ( u 2 $_{p}$v 2 k − u 2 $_{k}$v 2 $_{p}$)+ u 2 $_{k}$v 2 p and that the combination in braces vanishes in the weak coupling as it leads to an integral of the type (23). The expression for Π$_{FG}$( q ) above applies at arbitrary couplings; however Eq. (14) presumes weak-coupling ap- proximation, because Π$_{GF}$ ( q ) = − Π$_{FG}$( q ) holds only in this limit. The weak-coupling limit for the func- tion Π$_{FG}$( q ) is obtained on substituting in square braces u 2 k = u 2 p = v 2 p = v 2 k = 1 / 2. This statement is equivalent to ignoring integrals of the type The contributions to the polarization function due to quasiparticle scattering and pair breaking separate. For the contributions from scattering the poles are located at ± ( ϵ$_{p}$ − ϵ$_{k}$ ) and the distribution is given by the combina- tion f ( ϵ$_{p}$ ) − f ( ϵ$_{k}$ ). For the pair breaking contributions the poles are located at ± ( ϵ$_{p}$ + ϵ$_{k}$ ) and the distribution is pro- portional 1 − f ( ϵ$_{p}$ ) − f ( ϵ$_{k}$ ). The pair breaking contribu- tion vanishes in the limit T → T − c ( T is the temperature, T$_{c}$ is the critical temperature of the phase transition). The integral vanishes because ξ$_{p}$ changes sign for mo- menta above and below the Fermi momentum, while the remainder of the integrand is an even function in the vicinity of p$_{F}$ . Thus the loop polarization function re- duces to Now we write the retarded polarization functions Π $^{R}$( q ) in terms of the functions A ( q ), B ( q ) and C ( q ) af- ter performing analytical continuation ( iq → ω + iδ ) in functions Π$_{GG}$( q ), Π$_{FF}$ ( q ), Π$_{FG}$( q ) Π$_{G}$ † $_{G}$( q ). After some straightforward algebraic transformations we find