$\mathbf{c_\textbf{SW}}$ at One-Loop Order for Brillouin Fermions

TL;DR

This paper computes the one-loop correction to the clover coefficient for Brillouin fermions, showing it differs significantly from Wilson fermions, which impacts lattice QCD improvement strategies.

Contribution

The authors derive Feynman rules and calculate the one-loop ${c_ ext{SW}}$ coefficient for Brillouin fermions, providing a key comparison to Wilson fermions.

Findings

${c_ ext{SW}}^{(1)}$ for Brillouin fermions is approximately 0.124

${c_ ext{SW}}^{(1)}$ for Wilson fermions is approximately 0.269

Brillouin fermions have a smaller one-loop correction than Wilson 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}}^{(1)}_\mathrm{Brillouin} =0.12362580(1) $, compared to ${c_\mathrm{SW}}^{(1)}_\mathrm{Wilson} = 0.26858825(1)$, both for $r=1$.