content_image/416609.jpg content_image/416610.jpg content_image/416611.jpg content_image/416612.jpg content_image/416613.jpg content_image/416614.jpg content_image/416615.jpg content_image/416608.jpg
where ν ( p$_{F}$ ) = gm $^{∗}$p$_{F}$ / 2 π 2 is the density of states at the Fermi surface, and 

<img src='content_image/416609.jpg'>

<img src='content_image/416610.jpg'>

<img src='content_image/416611.jpg'>

These integrals are evaluated in Appendix B. We obtain the following analytical result for the imaginary part of P $_{2}$: 

<img src='content_image/416612.jpg'>

To leading-order the imaginary part of the polarization tensor is then given by 

<img src='content_image/416613.jpg'>

The real part of the polarization tensor follows from the dispersion (Kramers-Kronig) relation: 

<img src='content_image/416614.jpg'>

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 

<img src='content_image/416615.jpg'>

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 ). 

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 

<img src='content_image/416608.jpg'>

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 ). 

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. 

## 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 

