k}$ and ${\bm p}$ in addition to ${\bm q}$. Such a structure is generated in perturbation theory from the bare theory, where the interaction amplitude is simply given by the gauge field propagator, as we demonstrate in Appendix \[app:D\]. We then have S\_ &=& ()\^2 \_[q,k,p]{} (\_[\_1 \_4]{}\_[\_3 \_2]{})\
&& \_[\_1]{}(k-q/2) [|]{}\_[\_3]{}(p+q/2) \_[\_4]{}(p-q/2) \_[\_2]{}(k+q/2). \[eq:3.8\] Hermiticity requires ${\cal D}^*_{{\bm k},{\bm p}}({\bm q}) = {\cal D}_{{\bm
k},{\bm p}}(-{\bm q})$ (see also Eqs. (\[eqs:B.3\])). We now employ the phase space decomposition explained in Appendix \[app:B\] and focus on the large-angle scattering term, Eq. (\[eq:B.4b\]). Projecting again on the spin density we obtain \[eqs:3.9\] S\_ = \_q [D]{}([q]{}) ([n]{}\_(q)\_(-q)), \[eq:3.9a\] where ([q]{}) &=& \_[[k]{},[p]{}]{} g\_[k]{}g\_[p]{} ([k]{})\
&&+ \_[[k]{},[p]{}]{} g\_[k]{}g\_[p]{} ([k]{}-[p]{})\
&& [D]{}\^[(1)]{}([q]{}) + [D]{}\^[(2)]{}([q]{}). \[eq:3.9b\]

Here g\_[k]{} = (\_[k]{} - ) \[eq:3.10\] is a function that results from the projection onto the spin density and pins ${\bm k}$ and ${\bm p}$ to the Fermi surface.[@projection_footnote] Equation (\[eq:3.9a\]) has the form of the result obtained by Moriya.[@Moriya_1960] In what follows, we discuss the nature of the vector ${\bm D}({\bm q})$ in some more detail.

${\bm D}^{(2)}$ has the form \[eqs:3.11\] \^[(2)]{}([q]{}) = i [q]{}\^[(2)]{} + O([q]{}\^3), \[eq:3.11a\] with ${\bm d}^{\,(2)}$ a real vector given 