Gradient flow and the renormalization group

We investigate the renormalization group (RG) structure of the gradient flow. Instead of using the original bare action to generate the flow, we propose to use the effective action at each flow time. We write down the basic equation for scalar field theory that determines the evolution of the action, and argue that the equation can be regarded as an RG equation if one makes a field-variable transformation at every step such that the kinetic term is kept in the canonical form. We consider a local potential approximation (LPA) to our equation, and show that the result has a natural interpretation with Feynman diagrams. We make an epsilon expansion of the LPA and show that it reproduces the eigenvalues of the linearized RG transformation around both the Gaussian and the Wilson-Fisher fixed points to the order of epsilon.