Calculation of $c_\mathrm{SW}$ at one-loop order for Brillouin fermions

The Brillouin action is a Wilson-like lattice fermion action with a 81-point stencil, which was found to ameliorate the Wilson action in many respects. The Sheikholeslami-Wohlert coefficient $c_\mathrm{SW}$ of the clover improvement term has a perturbative expansion $c_\mathrm{SW}=c_\mathrm{SW}^{(0)}+g_0^2c_\mathrm{SW}^{(1)}+\mathcal{O}(g_0^4)$. At tree-level $c_\mathrm{SW}^{(0)}=r$ holds for Wilson and Brillouin fermions alike. We present the Feynman rules for the Brillouin action in lattice perturbation theory, and employ them to calculate the one-loop coefficient $c_\mathrm{SW}^{(1)}$ with plaquette or L\"uscher-Weisz gluons. Numerically its value is found to be about half that of the Wilson action.