Response functions of cold neutron matter: density fluctuations Armen Sedrakian and Jochen Keller Institute for Theoretical Physics, J. W. Goethe University, D-60438 Frankfurt-Main, Germany (Dated: October 26, 2018) We compute the finite temperature density response function of nonrelativistic cold fermions with an isotropic condensate. The pair-breaking contribution to the response function evaluated in the limit of small three-momentum transfers q within an effective theory which exploits series expansion in powers of small q/p F , where p F is the Fermi momentum. The leading order O ( q $^{2}$) contribution is universal and depends only on two fundamental scales, the Fermi energy and the pairing gap. The particle-hole Landau Fermi-liquid interaction contributes first at the next-to-leading order O ( q $^{4}$). The scattering contribution to the polarization tensor is nonperturbative (in the above sense) and is evaluated numerically. The spectral functions of density fluctuations are constructed and the relevance of the q 2 scaling for the pair breaking neutrino emission from neutron stars is discussed. I. INTRODUCTION The long-wavelength, low-energy dynamics of fermionic systems is determined by their response functions to soft perturbations, which are characterized by length scales that are large compared to the inverse Fermi wave vector and energies that are small compared to the Fermi energy. At zero temperature the response functions of pair-correlated nuclear matter have been studied long ago by Larkin and Migdal [1]. Recently, the response functions of pair-correlated nuclear systems at nonzero temperature received attention in the context of neutral current neutrino emission via pair breaking and formation in compact stars [2–5] and neutrino scattering in supernovae [6]. The evaluation of the response func- tions involves typically a resummation of infinite number of finite temperature ring diagrams. In the unpaired limit of normal Fermi liquid, this resummation scheme reduced to the familiar random-phase-approximation (RPA); for recent applications in nucleonic and neutron star matter, see Refs. [7–10]. In attractive, cold, fermionic systems the gap Δ in the quasiparticle spectrum is small compared to the Fermi energy and the hierarchy of energy scales depends on the magnitude of the perturbation, which can take arbi- trary values with respect to the pairing gap Δ. In this work we focus on density perturbations and show that the two distinct contributions to the response function through the scattering and pair breaking processes are effective below and above the energy threshold 2Δ, i.e., the energy needed to break a pair. The pair breaking processes are of special importance for applications in compact stars; to evaluate them, we propose a new sys- tematic low-transferred-momentum expansion of the re- sponse function which builds on the previous work on po- larization tensors of cold superfluid fermionic systems [2]. Specifically, we show that the pair breaking contribution to the polarization tensor posses a well-defined expansion with respect to the ratio of the momentum of the external current to the Fermi momentum of the fermions. We also adopt more general ansatz for the driving terms in the integral equations of Ref. [2] by lifting the degeneracy among the particle-particle and particle-hole channels. This paper is organized as follows. In Sec. II we de- rive the vertex functions and polarization tensor in a more general setting that in Ref. [2] by using different particle-particle and particle-hole interactions and clar- ify the approximations that arise in the weak coupling Bardeen-Cooper-Schrieffer (BCS) limit. A small momen- tum transfer expansion is applied to the pair breaking po- larization tensor in Sec. III A. We also show in Sec. III B It is now well established that at zeroth order ( q = 0) the pair breaking density response function vanishes, as required by the f -sum rule for the polarization tensor, which is a direct consequence of the baryon number con- servation. Some authors found analytically the leading- order contribution to the polarization which arises at or- der q 4 [3, 18]. Since there is no general argument that requires the coefficient of the leading-order q 2 term to be zero, the answer may depend on the approximations in- volved in the theory. In section III A we study in detail a new small q expansion of the polarization tensor obtained in Ref. [2] and find that the coefficient of the q 2 term in the series expansion of the density response function is indeed nonzero. Furthermore, it turns out to be univer- sal, i.e., it depends only on two fundamental scales of the problem, the Fermi energy and pairing gap, and is independent of the strength of particle-hole interaction. The density response functions can be utilized to deter- mine the spectrum of the collective modes, the stability of the system toward clustering, the rates of electromag- netic and weak radiation processes, and so on. In partic- ular, the rates of neutrino reactions in stellar interiors can be expressed through the response function of underly- ing matter to vector and axial-vector weak currents [11– 14]. In nonrelativistic limit, the vector and axial-vector responses are mapped onto the responses to the den- sity and spin-density perturbations, respectively. The response functions in the superfluid neutron matter were recently computed and the neutrino emission rates were determined in Refs. [2–5]. Phenomenologically, these are important in modeling the cooling of intermediate age neutron stars and the superburst in accreting neutron stars [15–17]. We work in the nonrelativistic limit, i.e., the ratio of the Fermi velocity to the speed of light is small ( v$_{F}$ /c ≪ 1). arXiv:1001.0395v2 [nucl-th] 19 Apr 2010