${\mathbf{c}}_{\mathbf{\text{SW}}}$ at One-Loop Order for Brillouin Fermions

Wilson-like Dirac operators can be written in the form $D={\gamma }_{\mu }{\mathrm{\nabla }}_{\mu }-\frac{ar}{2}\mathrm{\Delta }$. For Wilson fermions the standard two-point derivative ${\mathrm{\nabla }}_{\mu }^{(\mathrm{s}\mathrm{t}\mathrm{d})}$ and 9-point Laplacian ${\mathrm{\Delta }}^{(\mathrm{s}\mathrm{t}\mathrm{d})}$ are used. For Brillouin fermions these are replaced by improved discretizations ${\mathrm{\nabla }}_{\mu }^{(\mathrm{i}\mathrm{s}\mathrm{o})}$ and ${\mathrm{\Delta }}^{(\mathrm{b}\mathrm{r}\mathrm{i})}$ 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 $c_{\mathrm{S}\mathrm{W}}$, which, similar to the Wilson case, has a perturbative expansion of the form $c_{\mathrm{S}\mathrm{W}}=1+{c_{\mathrm{S}\mathrm{W}}}^{(1)}{g}_0^2+\mathcal{O}(g_0^4)$. For $N_c=3$ we find ${c_{\mathrm{S}\mathrm{W}}}_{\mathrm{B}\mathrm{r}\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{u}\mathrm{i}\mathrm{n}}^{(1)}=0.12362580(1)$, compared to ${c_{\mathrm{S}\mathrm{W}}}_{\mathrm{W}\mathrm{i}\mathrm{l}\mathrm{s}\mathrm{o}\mathrm{n}}^{(1)}=0.26858825(1)$, both for $r=1$.