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Lattice gradient flow with tree-level $\mathcal{O}(a^4)$ improvement in pure Yang-Mills theory

Following a recent paper by Fodor et al. (arXiv:1406.0827), we reexamine several types of tree-level improvements on the flow action with various gauge actions in order to reduce the lattice discretization errors in the Yang-Mills gradient flow method. We propose two types of tree-level, $\mathcal{O}(a^4)$ improved lattice gradient flow including the rectangle term in both the flow and gauge action within the minimal way. We then perform numerical simulations with the simple plaquette gauge action for testing our proposal. Our numerical results of the expectation value of the action density, $\langle E(t)\rangle$, show that two $\mathcal{O}(a^4)$ improved flows significantly eliminate the discretization corrections in the small flow time $t$ regime. On the other hand, the values of $t^2\langle E(t)\rangle$ in the large $t$ regime, where the lattice spacing dependence of the tree-level term dies out as inverse powers of $t/a^2$, are different between the results given by two optimal flows leading to the same $\mathcal{O}(a^4)$ improvement at tree level. This may suggest that non-negligible $\mathcal{O}(g^2 a^2)$ effect sets in the large $t$ regime, where the running coupling $g(1/\sqrt{8t})$ becomes large.