One-loop $c_\mathrm{SW}$ for Wilson and Brillouin fermions with stout smearing or Wilson flow

TL;DR

This work extends the one-loop determination of the Clover coefficient $c_ ext{SW}^{(1)}$ to Wilson and Brillouin fermions with either stout smearing or Wilson flow, on both plaquette and Symanzik gauge actions. By applying an upgrade recipe to perturbative calculations, the authors show that smoothing improves the perturbative series and that $c_ ext{SW}^{(1)}$ becomes close to its tree-level value $c_ ext{SW}^{(0)}=r$ for moderate couplings $g_0^2$, especially when $n_ ext{stout} ho$ or $t/a^2$ are sizable. The results, decomposed into $N_c$ and $1/N_c$ parts, indicate substantial reductions in $c_ ext{SW}^{(1)}$ with Brillouin fermions and with improved gluon actions, and validate a near-equivalence between stout smearing and Wilson flow. These findings support using the computed one-loop values for $c_ ext{SW}$ in practical simulations with smoothing to achieve better continuum behavior. The work provides a concrete, cross-checked framework for incorporating smoothing effects into lattice perturbation theory and offers actionable guidance for tuning $c_ ext{SW}$ in smoothed-link lattice QCD.

Abstract

We present results for the one-loop value of the improvement coefficient $c_\mathrm{SW}$ for Wilson and Brillouin fermions subject to stout smearing or Wilson flow, in combination with Wilson or Symanzik glue. To this end we use a recently developed method that allows one to modify an existing perturbative calculation, like the one for $c_\mathrm{SW}^{(1)}$, to include stout smearing or Wilson flow at arbitrary stout parameters ($\varrho$, $n_\mathrm{stout}$) or flow times $t/a^2$, respectively. Our results indicate that already a small amount of smoothing makes the perturbative series well behaved, suggesting that a non-perturbatively determined $c_\mathrm{SW}$ might be close to its one-loop value for couplings $g_0^2\simeq 1$.

Figures (25)

• Figure 1: Momentum assignments for the vertices with one, two and three gluons.
• Figure 2: The six one-loop diagrams contributing to the vertex function.
• Figure 3: The one-loop improvement coefficient $c_\mathrm{SW}^{(1)}$ with $N_c=3$ and up to four steps of stout smearing as a function of the smearing parameter $\varrho$. The four panels reflect the four combinations of the Wilson or Brillouin fermion (left vs. right) with plaquette or Lüscher-Weisz glue (top vs. bottom).
• Figure 4: Same as in Figure \ref{['fig:csw_stout']} but for the $c_\mathrm{SW}^{(1)}$ contribution linear in $N_c$.
• Figure 5: Same as in Figure \ref{['fig:csw_stout']} but for the $c_\mathrm{SW}^{(1)}$ contribution linear in $1/N_c$.