6 0 1 2 3 4 5 -0.006 -0.004 -0.002 0 Re  , Im  T = 0.2 T$_{c}$ T = 0.9 T$_{c}$ I$_{0}$ = 4 ν ( p$_{F}$ ) − $^{1}$µ ∫ d $^{3}$p (2 π ) 3 ξ$_{p}$ ϵ$_{p}$ω 2 [ 1 + ω 2 4 ϵ 2 p ( 6 µξ$_{p}$ ϵ 2 p x 2 − 1 ) ] L$_{0}$, (39) I$_{1}$ = 2 ν ( p$_{F}$ ) − $^{1}$µ ∫ d $^{3}$p (2 π ) 3 ξ$_{p}$ ϵ 3 p [ 4 ϵ 2 p ω 2 − 1 ] L$_{1}$ x $^{2}$, (40) I$_{2}$ = 2 ν ( p$_{F}$ ) − 1 ∫ d $^{3}$p (2 π ) 3 1 ϵ$_{p}$ [ 4 ϵ 2 p ω 2 − 1 ] L$_{2}$. (41) Im P 2 = − πν ( p$_{F}$ ) 3 T Δ $^{2}$µ sgn( ω ) ω 5 √$_{ω}$ 2 − 4Δ 2 sech 2 ( ω 4 T ) { 4 µω ( ω 2 − 4Δ $^{2}$) − T [40Δ $^{2}$µ − 3 ω $^{2}$(6 µ + √ ω 2 − 4Δ $^{2}$)] sinh ( ω 2 T ) } θ ( | ω | − 2Δ) . (42) ImΠ $^{R}$( q ) = Im P $_{2}$( ω ) ( q p$_{F}$ ) 2 + O ( q $^{4}$) . (43) ReΠ $^{R}$( ω, q ) = − 1 π ∫ dω $_{′}$ImΠ $^{R}$( ω $^{′}$, q ) ω − ω ′ . (44) B ( ω, q ) = − 2ImΠ $^{R}$( ω, q ) [ ω 2 − q 2 − ReΠ $^{R}$( ω, q )] 2 + ImΠ $^{R}$( ω, q ) $_{2}$. (45) where ν ( p$_{F}$ ) = gm $^{∗}$p$_{F}$ / 2 π 2 is the density of states at the Fermi surface, and These integrals are evaluated in Appendix B. We obtain the following analytical result for the imaginary part of P $_{2}$: FIG. 1: Dependence of the real (solid lines) and imaginary (dashed lines) parts of the pair breaking polarization ten- sor in units of density of states on the energy transfer in units of threshold energy 2Δ( T ) for fixed momentum trans- fer q = 0 . 1 p F , with p F = 0 . 1 fm − $^{1}$, and two tempera- tures T = 0 . 2 T c (heavy lines) and T = 0 . 9 T c (light lines). The zero temperature gap is taken to be Δ(0) = 1 . MeV, T c = Δ(0) / 1 . 76. To leading-order the imaginary part of the polarization tensor is then given by frequency 2Δ(0), the momentum transfer to the Fermi momentum. The real and imaginary parts of the polar- ization tensor scale as q $^{2}$. Their behavior at negative en- ergies follows from their even and odd parity with respect to the energy transfer, i.e., ReΠ $^{R}$( − ω ) = ReΠ $^{R}$( ω ) and ImΠ $^{R}$( − ω ) = − ImΠ $^{R}$( ω ). Note that the imaginary parts are identically zero below the threshold for pair breaking process 2Δ( T ). The real part of the polarization tensor follows from the dispersion (Kramers-Kronig) relation: Figure 2 illustrates the spectral functions of pair break- ing density fluctuation on the energy and momentum transfers. The parameters are the same as in Fig. 1. The form of the spectral function suggests that at low temper- atures the low-momentum-transfer contribution is con- centrated near the pair breaking threshold; for larger mo- mentum transfers, modes away from the energy threshold become important. At higher temperatures ( T ≤ T$_{c}$ ) and for any given momentum transfer, the main contribution to the spectral function comes from higher energy modes and the peak values are larger in the latter regime. Using these quantities one can construct an effective the- ory of collective excitations. Their (full, interacting) propagators are completely determined by the spectral function of the collective excitations The dispersion relation of the collective excitations is read-off as ω 2 = q 2 + ReΠ $^{R}$( ω, q ). The finite life-time effects are described by the width of the spectral func- tion, i.e., by the function ImΠ $^{R}$( ω, q ). B. Scattering response function: numerical evaluation The scattering part of the response function is kine- matically important for the space-like processes and van- ishes automatically in the limit q → 0. Small momentum expansion of the previous section was found inappropri- ate for the scattering part of the polarization function and it was evaluated numerically by adapting the method Figure 1 illustrates the dependence of the real and imaginary parts of the pair breaking polarization ten- sor on the transferred energy for fixed three-momentum transfer. The value of the Landau parameter is f$_{0}$ = − 0 . 5 [2]. The zero temperature gap at p$_{F}$ = 0 . 1 fm − 1 is Δ(0) = 1 MeV and T$_{c}$ = Δ(0) / 1 . 76. The frequen- cies are normalized to the zero-temperature threshold