by \^[(2)]{} &=& \_[[k]{},[p]{}]{} g\_[k]{}g\_[p]{} ([p]{}-[k]{}) . \[eq:3.11b\]

${\bm D}^{(1)}$ can be written \[eqs:3.12\] D\^[(1)]{}\_i([q]{}) = i D\_[ij]{}q\_j + O(q\^3), \[eq:3.12a\] with $D_{ij}$ a real rank-2 tensor given by D\_[ij]{} = \_[[k]{},[p]{}]{} g\_[k]{}g\_[p]{}\_[ilm]{}k\_l p\_m[E]{}\_j([k]{},[p]{}), \[eq:3.12b\] where \_i([k]{},[p]{}) &=& d[x]{}d[y]{}(x\_i-y\_i) e\^[-i([k]{}+[p]{})([x]{}+[y]{})/2]{}\
&&-20pt d[z]{} (([k]{}-[p]{}))[D]{}([x]{},[y]{};[z]{}) \[eq:3.12c\] with ${\cal D}({\bm x},{\bm y};{\bm z})$ the Fourier transform of ${\cal
D}_{{\bm k},{\bm p}}({\bm q})$ in analogy to Eq. (\[eq:B.2b\]). $D_{ij}$ has a symmetric part and an antisymmetric part. The latter can be combined with ${\bm D}^{(2)}$ above to form a contribution to ${\bm D}$ that we denote by \[eqs:3.13\] \^[(-)]{}([q]{}) = i[q]{} + O([q]{}\^3), \[eq:3.13a\] where d\_i = d\_i\^[(2)]{} + \_[[k]{},[p]{}]{} g\_[k]{}g\_[p]{} (p\_i k\_j - k\_i p\_j)[E]{}\_j([k]{},[p]{}). \[eq:3.13b\] The symmetric part can be written as a diagonal tensor plus a traceless rank-2 tensor, and the latter can be diagonalized by means of an orthogonal transformation that amounts to a spatial rotation. The symmetric part of $D_{ij}$ we thus can write \[eqs:3.14\] D\^[(+)]{}\_[ij]{} = D \_[ij]{} + a\_i\_[ij]{} \[eq:3.14a\] ([*no*]{} summation convention) where the $a_i$ obey \_i a\_i = 0. \[eq:3.14b\] If desirable, the $a_i$ can be explicitly constructed from Eq.(\[eq:3.12b\]). We thus have a second contribution to ${\bm D}$ that we denote by \^[(+)]{}([q]{}) = 