Topological charge using cooling and the gradient flow

Abstract

The equivalence of cooling to the gradient flow when the cooling step ${n}_{c}$ and the continuous flow step of gradient flow $\ensuremath{\tau}$ are matched is generalized to gauge actions that include rectangular terms. By expanding the link variables up to subleading terms in perturbation theory, we relate ${n}_{c}$ and $\ensuremath{\tau}$ and show that the results for the topological charge become equivalent when rescaling $\ensuremath{\tau}\ensuremath{\simeq}{n}_{c}/(3\ensuremath{-}15{c}_{1})$, where ${c}_{1}$ is the Symanzik coefficient multiplying the rectangular term. We, subsequently, apply cooling and the gradient flow using the Wilson, the Symanzik tree-level improved, and the Iwasaki gauge actions to configurations produced with ${N}_{f}=2+1+1$ twisted mass fermions. We compute the topological charge, its distribution, and the correlators between cooling and gradient flow at three values of the lattice spacing demonstrating that the perturbative rescaling $\ensuremath{\tau}\ensuremath{\simeq}{n}_{c}/(3\ensuremath{-}15{c}_{1})$ leads to equivalent results.