5 A ( q ) = ∫ d $^{3}$p (2 π ) 3 { 1 2 ( ϵ$_{k}$ − ϵ$_{p}$ ) [ 1 + ξ$_{p}$ξ$_{k}$ ϵ$_{p}$ϵ$_{k}$ − Δ 2 ϵ$_{p}$ϵ$_{k}$ ] f ( ϵ$_{p}$ ) − f ( ϵ$_{k}$ ) ( ω + iδ ) 2 ( ϵ$_{p}$ − ϵ$_{k}$ ) 2 + 1 2 ( ϵ$_{p}$ + ϵ$_{k}$ ) [ 1 − ξ$_{p}$ξ$_{k}$ ϵ$_{p}$ϵ$_{k}$ + Δ 2 ϵ$_{p}$ϵ$_{k}$ ] f ( − ϵ$_{p}$ ) − f ( ϵ$_{k}$ ) ( ω + iδ ) 2 ( ϵ$_{p}$ + ϵ$_{k}$ ) 2 } , (25) B ( q ) = − Δ ω ∫ d $^{3}$p (2 π ) 3 1 ϵ$_{p}$ [ f ( ϵ$_{p}$ ) − f ( ϵ$_{k}$ ) ( ω + iδ ) 2 ( ϵ$_{p}$ − ϵ$_{k}$ ) 2 − f ( − ϵ$_{p}$ ) − f ( ϵ$_{k}$ ) ( ω + iδ ) 2 ( ϵ$_{p}$ + ϵ$_{k}$ ) 2 ] , (26) C ( q ) = 1 2 ∫ d $^{3}$p (2 π ) 3 { [ ( ϵ$_{p}$ − ϵ$_{k}$ ) ( 1 − ξ$_{p}$ξ$_{k}$ ϵ$_{p}$ϵ$_{k}$ − Δ 2 ϵ$_{p}$ϵ$_{k}$ )] f ( ϵ$_{p}$ ) − f ( ϵ$_{k}$ ) ( ω + iδ ) 2 ( ϵ$_{p}$ − ϵ$_{k}$ ) 2 − [ ( ϵ$_{p}$ + ϵ$_{k}$ ) ( 1 + ξ$_{p}$ξ$_{k}$ ϵ$_{p}$ϵ$_{k}$ + Δ 2 ϵ$_{p}$ϵ$_{k}$ )] f ( − ϵ$_{p}$ ) − f ( ϵ$_{k}$ ) ( ω + iδ ) 2 ( ϵ$_{p}$ + ϵ$_{k}$ ) 2 − 1 ϵ$_{p}$ [1 − 2 f ( ϵ$_{p}$ )] } . (27) Note that lim$_{q}$$_{→}$$_{0}$$_{,ω}$$_{→}$$_{0}$ C ( q ) = 0, since the coupling con- stant in the particle-particle channel can be expressed as ( v$_{pp}$ ) − 1 = Π$_{G}$ † $_{G}$( q = 0) + Π$_{FF}$ ( q = 0) = ∫ d $^{3}$p (2 π ) 3 1 2 ϵ$_{p}$ [1 − 2 f ( ϵ$_{p}$ )] , (28) which is the gap equation for the contact interaction v$_{pp}$ . We do not need to specify the regularization of the gap equation, since its divergence is eliminated in the loop integrals. For numerical purposes we will adopt gaps obtained from finite-range interactions in Ref. [19]. III. EVALUATING RESPONSE FUNCTIONS Equations (25)–(27) separate into scattering and pair breaking contributions. We shall see that the first con- tributes essentially below the pair breaking threshold ω < 2Δ, whereas the second contributes for ω > 2Δ. In the following we discuss in detail the pair breaking part and its small momentum expansion. The scatter- ing part will be addressed later in this section, where we evaluate it numerically. A. pair breaking response function: small momentum expansion The small- y expansion of the pair breaking part of the polarization tensor is obtained upon writing Π $^{R}$( q ) = ∑ n P $_{n}$y$^{n}$, A ( q ) = ∑ n A $_{n}$y n and similarly for the func- tions B and C and truncating the Taylor series at the de- sired order in y . The odd powers of y do not contribute to the series. Up to order y 4 the coefficients for the polarization ten- P 2 = Δ $^{2}$ν ( p$_{F}$ )( I$_{0}$ + I$_{1}$ + I$_{2}$ ) , (38) where the functions L$_{0}$ , L$_{1}$ , and L$_{2}$ are defined by Eqs. (A10)–(A12) of Appendix A. The zeroth order term P 0 vanishes as a consequence of the f -sum rule. The leading order nonzero term is given by sor are given by P 0 = 0 , (29) P 2 = 2 B 0 B 2 + A 2 C 0 + A 0 C 2 , (30) C 0 P 4 = B 2 + 2 B 0 B 4 + A 4 C 0 + A 2 C 2 + A 0 C 4 0 − P 2 [ C 2 0 − v$_{ph}$ P 2 ] . (31) The coefficients of the expansion are given by (for details see Appendix A) A 0 = 2Δ 2 ∫ d $^{3}$p (2 π ) 3 1 ϵ$_{p}$ L$_{0}$, (32) B 0 = Δ ω d $^{3}$p (2 π ) 3 1 ϵ$_{p}$ L$_{0}$, (33) C 0 = − ω 2 2 ∫ d $^{3}$p (2 π ) 3 1 ϵ$_{p}$ L$_{0}$, (34) A 2 = ∫ d $^{3}$p (2 π ) 3 [ Δ 2 µ ξ$_{p}$ ϵ 5 p (6 µξ$_{p}$x 2 − ϵ 2 $_{p}$) L$_{0}$ − 2Δ 2 µ ξ$_{p}$ ϵ 3 p L$_{1}$ x 2 + 2Δ 2 ϵ$_{p}$ L$_{2}$ ] , (35) B 2 = Δ ω ∫ d $^{3}$p (2 π ) 3 1 ϵ$_{p}$ L$_{2}$, (36) C 2 = − ∫ d $^{3}$p (2 π ) 3 [ µ ξ$_{p}$ ϵ$_{p}$ L$_{0}$ + 2 µξ$_{p}$ ϵ$_{p}$ L$_{1}$ x 2 + 2 ϵ$_{p}$L$_{2}$ ] , (37)