Determination of Reference Scales for Wilson Gauge Action from Yang--Mills Gradient Flow

A parametrization of the lattice spacing ($a$) in terms of the bare coupling ($\beta$) for the SU(3) Yang--Mills theory with the Wilson gauge action is given in a wide range of~$\beta$. The Yang--Mills gradient flow with respect to the flow time~$t$ for the dimensionless observable, $t\frac{d}{dt}t^2\langle E(t)\rangle$, is utilized to determine the parametrization. With fine lattice spacings ($6.3\le\beta\le7.5$) and large lattice volumes ($N_{\rm s}=64$--$128$), the discretization and finite-volume errors are significantly reduced to the same level as the statistical error.