Calculation of $c_{\mathrm{S}\mathrm{W}}$ 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 $c_{\mathrm{S}\mathrm{W}}$ of the clover improvement term has a perturbative expansion $c_{\mathrm{S}\mathrm{W}}=c_{\mathrm{S}\mathrm{W}}^{(0)}+g_0^2{c}_{\mathrm{S}\mathrm{W}}^{(1)}+\mathcal{O}(g_0^4)$. At tree-level $c_{\mathrm{S}\mathrm{W}}^{(0)}=r$ 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 $c_{\mathrm{S}\mathrm{W}}^{(1)}$ with plaquette or Lüscher-Weisz gluons. Numerically its value is found to be about half that of the Wilson action.