by a renormalization of the action ${\cal F}$, as we now proceed to show.

A renormalization of the action ${\cal F}$ generates additional terms by virtue of the anisotropic $M$-propagator, which now reads M\_i([k]{})M\_j(-[k]{})= +  . \[eq:2.5\] For instance, two-loop diagrams of the structure shown in Fig. \[fig:2.1\]

-0mm ![A diagram that generates a $({\bm\nabla}\cdot{\bm M})^2$ term.[]{data-label="fig:2.1"}](fig_2.1.eps "fig:"){width="3.0cm"}

both renormalize $d_{ij}({\bm k})$ and lead to a new vertex \_[k]{} k\_ik\_jM\_i([k]{})M\_j(-[k]{}), \[eq:2.6\] as well as to higher order anisotropic gradient terms. Equation (\[eq:2.6\]) represents the $({\bm\nabla}\cdot{\bm M})^2$ term that was mentioned above. Conversely, if one starts with a theory that contains a $({\bm\nabla}\cdot{\bm
M})^2$ term, which is allowed by symmetry and hence should be included in any Landau theory, then the leading dipole-dipole term will be generated in perturbation theory even if it was not included in the bare action. The complete LGW action for a classical, isotropic Heisenberg ferromagnet, up to terms quadratic in gradients and quartic in the order parameter, thus reads \[eqs:2.7\] &=& \_V d[x]{} \
&& - \_[V]{} d[x]{}d[y]{} M\_i([x]{}) d\_[ij]{}([x]{}-[y]{}) M\_j([y]{})\
&& + \_V d[x]{} (([x]{}))\^2. \[eq:2.7a\] Here $d_{ij}({\bm x}-{\bm y})$ is the Fourier transform of $d_{ij}({\bm k})$ in Eq. (\[eq:2.4b\]), namely, d\_[ij]{}([x]{}-[y]{}) =  . \[eq:2.7b\] 