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

Abstract

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 ${c_{\mathrm{SW}}}_{\mathrm{Brillouin}}^{(1)}=0.12362580(1)$, compared to ${c_{\mathrm{SW}}}_{\mathrm{Wilson}}^{(1)}=0.26858825(1)$, both for $r=1$.

Comments: Proceedings of the 39th International Symposium on Lattice Field Theory, 8th-13th August, 2022, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany