The Yang-Mills gradient flow and lattice effective action

Recently, the Yang-Mills gradient flow has been found to be a useful concept not only in lattice simulations but also in continuous field theories. Since its smearing property is similar to the Wilsonian "block spin transformation", there might be a deeper connection between them. In this work, we define the "effective action", which generates configurations at a finite flow time, and derive the exact differential equation to investigate the flow-time dependence of the action. Then the Yang-Mills gradient flow can be regarded as the flow of the effective action. We also propose a flow-time-dependent gradient, where the differential equation becomes similar to the renormalization group equation. We discuss the possibility of regarding the time evolution of the effective action as the Wilsonian renormalization group flow.