Gradient flow and the Wilsonian renormalization group flow

Abstract

The gradient flow is the evolution of fields and physical quantities along a dimensionful parameter t, the flow time. We give a simple argument that relates this gradient flow and the Wilsonian renormalization group (RG) flow. We then illustrate the Wilsonian RG flow on the basis of the gradient flow in two examples that possess an infrared fixed point, the 4D many-flavor gauge theory and the 3D O(N ) linear sigma model.

Results and techniques for higher order calculations within the gradient-flow formalism

“…The NLO term has been obtained in Ref. [54], the NNLO term is new. Again, only four decimal places are displayed and our uncertainty estimate for the numerical integration is several digits beyond that.…”

Derivation of a gradient flow from the exact renormalization group

“…The similarity between the gradient flow and the renormalization group flow was pointed out already at the beginning [2] and has been pursued further [8][9][10][11][12][13][14][15]. The purpose of this paper is to establish a concrete correspondence between the two flows for a generic real scalar field theory in D-dimensional Euclidean space.…”

“…The large t behavior of the left-hand sides have been calculated explicitly in the large N limit of the O(N) linear sigma model in D = 3 [13,14].…”

Stochastic renormalization group and gradient flow

“…Inspired by this resemblance, work has been done on the lattice to use gradient flow to define a continuous blocking transformation with which anomalous dimensions have been computed [3,4], with encouraging results. On the analytic side, work has been done [5][6][7] connecting GF to the framework of functional (or exact) RG (FRG), which is a method by which one defines RG transformations nonperturbatively and continuously in the context of continuum field theory [8][9][10][11]. 1 In particular, it has been noted that certain definitions of a GF effective action lead to a kind of Langevin equation [6], and most recently, that the connected n−point functions of a particular FRG effective theory are equal to the GF observables up to proportionality [7].…”