  Wilson-like Dirac operators can be written in the form
$D=\gamma_\mu\nabla_\mu-\frac {ar}{2} \Delta$. For Wilson fermions the standard
two-point derivative $\nabla_\mu^{(\mathrm{std})}$ and 9-point Laplacian
$\Delta^{(\mathrm{std})}$ are used. For Brillouin fermions these are replaced
by improved discretizations $\nabla_\mu^{(\mathrm{iso})}$ and
$\Delta^{(\mathrm{bri})}$ which have 54- and 81-point stencils respectively. We
derive the Feynman rules in lattice perturbation theory for the Brillouin
action and apply them to the calculation of the improvement coefficient
${c_\mathrm{SW}}$, which, similar to the Wilson case, has a perturbative
expansion of the form
${c_\mathrm{SW}}=1+{c_\mathrm{SW}}^{(1)}g_0^2+\mathcal{O}(g_0^4)$. For $N_c=3$
we find ${c_\mathrm{SW}}^{(1)}_\mathrm{Brillouin} =0.12362580(1) $, compared to
${c_\mathrm{SW}}^{(1)}_\mathrm{Wilson} = 0.26858825(1)$, both for $r=1$.
