the Coulomb interaction that is a much larger effect than the relativistic dynamics inherent in Eqs. (\[eqs:3.2\]).

Equation (\[eq:3.3a\]) contains number density fluctuations at all wavelengths. If one restricts the theory to interactions between long-wavelength fluctuations, then this interaction can be rewritten as a sum of parts that includes an interaction between spin-density fluctuations, see Ref.  and Appendix \[app:B\]. The basic point is that an interaction between number density fluctuations at large wave numbers can be written as one between spin density fluctuations at small wave numbers. In an effective low-energy theory that contains only fluctuations at wave numbers smaller than some cutoff $\lambda$, Eq. (\[eq:3.3a\]) therefore contains a contribution S\_ = ’ dx [n]{}\_(x) \_(x), \[eq:3.4\] where the prime on the integral indicates that only the small-wave-number contributions (smaller than $\lambda$) to the spin density ${\bm n}_{\text{s}}$ are to be considered in order to avoid overcounting. As has been explained in Ref. , it is convenient to choose the cutoff $\lambda$ as a fixed fraction of the Thomas-Fermi screening wave number, and the spin-triplet interaction amplitude $\Gamma_{\text{t}}$ is a Fermi-surface average over $v_{\,\text{C}}({\bm k}-{\bm p})\,\Theta(\vert {\bm
k}-{\bm p}\vert - \lambda)$, with ${\bm k}$ and ${\bm p}$ pinned to the 