Numerical study of tree-level improved lattice gradient flows in pure Yang-Mills theory

We study several types of tree-level improvement in the Yang-Mills gradient flow method in order to reduce the lattice discretization errors in line with Fodor et al. [arXiv:1406.0827]. The tree-level $\mathcal{O}(a^2)$ improvement can be achieved in a simple manner, where an appropriate weighted average is computed between two definitions of the action density $\langle E(t)\rangle$ measured at every flow time $t$. We further develop the idea of achieving the tree-level $\mathcal{O}(a^4)$ improvement. For testing our proposal, we present numerical results for $\langle E(t) \rangle$ obtained on gauge configurations generated with the Wilson and Iwasaki gauge actions at three lattice spacings ($a\approx 0.1, 0.07,$ and 0.05 fm). Our results show that tree-level improved flows significantly eliminate the discretization corrections on $t^2\langle E(t)\rangle$ in the relatively small-$t$ regime. To demonstrate the feasibility of our tree-level improvement proposal, we also study the scaling behavior of the dimensionless combinations of the $\Lambda_{\overline{\textrm{MS}}}$ parameter and the new reference scale $t_X$, which is defined through $t_X^2\langle E(t_X)\rangle=X$ for the smaller $X$, e.g., $X= 0.15$. It is found that $\sqrt{t_{0.15}}\Lambda_{\overline{\textrm{MS}}}$ shows a nearly perfect scaling behavior as a function of $a^2$ regardless of the types of gauge action and flow, after tree-level improvement is achieved up to $\mathcal{O}(a^4)$. Further detailed study of the scaling behavior exposes the presence of the remnant $\mathcal{O}(g^{2n} a^2)$ corrections, which are beyond the tree level. Although our proposal is not enough to eliminate all $\mathcal{O}(a^2)$ effects, we show that the $\mathcal{O}(g^{2n} a^2)$ corrections can be well under control even by the simplest tree-level $\mathcal{O}(a^2)$ improved flow.