2 II. DENSITY RESPONSE FUNCTION A. Vertex functions ϵ$_{p}$ = √ ξ 2 p + Δ $^{2}$( p ) , (1) V pp ≃ v$_{pp}$ + v$_{pp}$ ( σ · σ $^{′}$) + . . . , (3) V ph v$_{ph}$ + v$_{ph}$ ( σ σ $^{′}$) + . . . , (4) ξ$_{p}$$_{+}$$_{q}$ = p 2 2 m ∗ ( 1 + 2 p · q p 2 + q 2 p 2 ) − µ, (2) ˆ Γ$_{1}$ = Γ$_{0}$ + v$_{ph}$ ( G Γ$_{1}$ G + ˆ F ˆ Γ$_{3}$ G + G ˆ Γ$_{2}$ ˆ F + ˆ F Γ$_{4}$ ˆ F ) , (5) ˆ Γ$_{2}$ = v$_{pp}$ ( G ˆ Γ$_{2}$ G † + ˆ F Γ$_{4}$ G † + G Γ$_{1}$ ˆ F + ˆ F ˆ Γ$_{3}$ ˆ F ) , (6) ˆ Γ$_{3}$ = v$_{pp}$ ( G † ˆ Γ$_{3}$ G + ˆ F Γ$_{1}$ G + G $^{†}$Γ$_{4}$ ˆ F + ˆ F ˆ Γ$_{2}$ ˆ F ) , (7) ˆ Γ$_{4}$ = Γ$_{0}$ + v$_{ph}$ ( G $^{†}$Γ$_{4}$ G † + ˆ F Γ$_{1}$ ˆ F + ˆ F ˆ Γ$_{2}$ G † + G † ˆ Γ$_{3}$ ˆ F ) . (8) Π$_{XX}$ ′ ( q ) = g ∫ d $^{4}$p (2 π ) $_{4}$X ( p ) X $^{′}$( p + q ) , (9) the results of exact numerical evaluation of the scattering part of the polarization tensor. Further in Sec. III C we verify that the unpaired and uncorrelated limits are re- covered from the scattering part. Our conclusions are collected in Sec. IV. Details of calculations are pre- sented in Appendices A and B. We use the natural units ℏ = c = 1 throughout and assume that the Boltzmann constant k$_{B}$ = 1. In this section we derive the general form of the density-response function of neutron matter at nonzero temperature. At densities below the saturation density neutron matter forms a $^{1}$S$_{0}$ pair condensate and can be described by the weak coupling limit of the BCS theory. This affects not only the approximations that are applied to the gap equation, but also the approximate relations between the loop integrals, as we discuss below. In the following we shall demonstrate that the pair break- ing part of the response function can be expanded sys- tematically with respect to the y parameter. Such expan- sion is thus valid for small three-momentum transfers, but arbitrary energy transfers. The couplings in the particle-particle ( pp ) and particle- hole ( ph ) channels, v$_{pp}$ and v$_{ph}$ , are assumed zero range; often their values are taken to be degenerate v$_{pp}$ = v$_{ph}$ and equal to the lowest order Landau parameter f$_{0}$ . We shall lift this approximation below by assuming v$_{pp}$ = v$_{ph}$ . The spectrum of paired neutrons is given by where ξ$_{p}$ = p $^{2}$/ 2 m ∗ − µ is the quasiparticle spectrum in the normal state, Δ( p ) is the energy gap, p is the three-momentum, m ∗ is the effective mass and µ is the chemical potential. For contact pairing interaction the gap is momentum independent, Δ( p ) ≡ Δ. The softness of the modes implies that their wave vec- tor | q | ≪ p$_{F}$ . Accordingly, we write We start with integral equations for the vertex func- tions and derive a (slight) generalization of their coun- terparts in Ref. [2] that distinguish the particle-particle and particle-hole interactions. These we write as sums of central and spin-spin interaction terms where the ellipses stand for the tensor and spin-orbit terms that are subdominant at relevant densities in neu- tron matter. The integral equations defining the scalar vertex, which we write in an operator form, are given by [1, 2] and consider the second and third terms in the bracket as small compared to unity, since p ≃ p$_{F}$ , where p$_{F}$ is the Fermi momentum. Thus, we may write ξ$_{p}$$_{+}$$_{q}$ ≃ ξ$_{p}$ + µ$_{0}$ (2 yx + y $^{2}$), where x = ( p · q ) / ( | p || q | ) and y = q/p$_{F}$ , whereby y ≪ 1. Here µ$_{0}$ = p 2 $_{F}$/ 2 m ∗ is the chemical potential at zero temperature; we shall drop the 0 index in the following. Several observations are in order: Here ˆ F = − iσ$_{y}$F , σ$_{y}$ is the second component of the Pauli matrix, Γ$_{0}$ = 1. When v$_{pp}$ = v$_{ph}$ , Eqs. (5)-(8) become identical to those of Ref. [2]. Let us now define the “elementary loop” as 1. If the expansion is carried out with respect to the small parameter δξ = ξ$_{p}$$_{+}$$_{q}$ − ξ$_{p}$ , as in Ref. [2], the power counting is not manifest. At the leading- order the terms which scale linearly in x drop on angle integration in symmetrical limits. The only nonzero contribution proportional to q 2 is then fur- nished by the recoil term. One needs to carry out the δξ expansion at least to second order to obtain all relevant terms that are of order q $^{2}$. 2. An alternative to the expansion with respect to δξ is the expansion with respect to the ratio y = | q | /p$_{F}$ . This expansion exploits the softness of the modes and applies both for timelike and spacelike momentum transfers. It is an alternative to ex- pansions in the ratios ω/qv$_{F}$ and qv$_{F}$ /ω , which are valid in these regimes, respectively. 3. Organizing the expansion in powers of v$_{F}$ /c ≪ 1 (nonrelativistic fermions) does not guarantee per se the convergence of the series. At any fixed density, the Fermi velocity v$_{F}$ is constant and for sufficiently large momentum transfers ( q ≥ p$_{F}$ / 2) the series will fail to converge. 4. Finally, the smallness of the expansion parameter is necessary but not sufficient condition for the con- vergence of the Taylor series. The validity of the expansion should be checked by an exact numerical computation of the loop integrals.