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Note that lim$_{q}$$_{→}$$_{0}$$_{,ω}$$_{→}$$_{0}$ C ( q ) = 0, since the coupling con- stant in the particle-particle channel can be expressed as 

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- 

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sor are given by 

The coefficients of the expansion are given by (for details see Appendix A) 

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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 

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