content_image/383265.jpg content_image/383274.jpg content_image/383275.jpg content_image/383273.jpg
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where the function A ( q ) in the unpaired state reduces to 

<img src='content_image/383274.jpg'>

Equations (48) and (49) are the standard expressions for the polarization tensor of a Fermi liquid in the random phase approximation (RPA). The free Fermi gas result follows on setting v$_{ph}$ = 0 in Eq. (48): 

Clearly, the pair-braking contribution vanishes as T → T ( − ) c . The scattering contribution at T ≥ T$_{c}$ reproduces the unpaired and uncorrelated limits, as it should. 

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In this article we have carried out several steps in the program aimed at understanding the polarization tensor of pair-correlated neutron and nuclear matter by (i) con- structing a low-transfered-momentum approximation to the pair breaking polarization tensor and (ii) by numer- ically evaluating the scattering part of the polarization tensor. 

Our main result is that the low-momentum expan- sion of the pair breaking polarization tensor starts at quadratic order in the ratio of the momentum-transfer to the Fermi momentum. The expansion coefficient is universal, i.e., depends only the two relevant scales: the Fermi energy and the pairing gap. We also clarified the structure of the integral equations for the vertex func- tions when the particle-particle and particle-hole inter- actions do not coincide and verified explicitly that the unpaired and uncorrelated limits are recovered. 

Further steps will require a verification of the con- vergence of the series by a comparison of our analyt- ical results with an exact numerical evaluation of the response functions. The rate of series convergence can be checked by computing the next-to-leading-order con- tribution. Further refinements could include finite range interactions, tensor forces, and so on. The new numerical and analytical methods, discussed in this article, could be useful in the studies of the response functions of pair- correlated fermionic systems in general. 

The main implication of our study concerns the neu- trino emissivity via the vector current pair breaking bremsstrahlung, which are phenomenologically impor- tant in the physics of neutron star cooling and super- bursts in accreting neutron stars [15–17]. Our results suggest that the vector current contribution to the pro- cess 

where { nn } refers to the pair correlated state, ν and ¯ ν to the neutrino and antineutrino, is suppressed to a lesser degree, than suggested in the recent literature [3, 18]. However, further studies indicated above are needed to draw a final conclusion on the relative importance of this vector current neutrino emission process (51). 

We thank T. Brauner, X.-G. Huang, and D. H. Rischke for discussions. The work of J. K. was supported by the Deutsche Forschungsgemeinschaft (Grant SE 1836/1-1). 

## Appendix A: Expanding the functions A ( q ) , B ( q ) , and C ( q ) 

We write each of these functions as a product of an ap- propriate coherence factor and corresponding statistical 

## IV. CONCLUSIONS 

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## Acknowledgments 

