Topological charge using cooling and the gradient flow

The equivalence of cooling to the gradient flow when the cooling step n(c) and the continuous flow step of gradient flow 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 tau and show that the results for the topological charge become equivalent when rescaling t similar or equal to n(c)/(3 - 15(c1)), 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 t similar or equal to n(c)/(3 - 15c(1)) leads to equivalent results.