Renormalization group properties of scalar field theories using gradient flow

Abstract

Gradient flow has proved useful in the definition and measurement of renormalized quantities on the lattice. Recently, the fact that it suppresses high-modes of the field has been used to construct new, continuous RG transformations both analytically and on the lattice, distinct from the usual blocking techniques in spin models and gauge theories. In this work, we discuss such a formulation for scalar field theory, and we present preliminary numerical results for its application to the determination of critical exponents at the Wilson-Fisher fixed point of three-dimensional φ 4 theory.

Stochastic Renormalization Group and Gradient Flow in Scalar Field Theory

“…3 In the case of φ − φ correlators, the formula actually predicts that the exponent of b ε (t) should be 0, meaning that correlations of φ operators cannot be used to determine γ φ . The correlator ratio should plateau at 1 at large distances z a t , and this was verified (see [6]). For the observable O = φ 2 , we expect to measure the difference δ 2 = 2(γ φ 2 − 2γ φ ) ≈ 0.752 [14].…”

“…In the past decade, a new tool known as "gradient flow" (GF) has become popular in lattice simulations as a means of setting the scale [1,2]. Because GF consists in a diffusion process on the degrees of freedom of the theory, which involves a local smoothing of the field, there have arisen attempts to characterize GF as a renormalization group transformation [3][4][5][6][7][8][9]. Since GF does not automatically involve a field rescaling, however, it is clear that it cannot in itself constitute an RG transformation (for non-compact fields, at least).…”

“…Since GF does not automatically involve a field rescaling, however, it is clear that it cannot in itself constitute an RG transformation (for non-compact fields, at least). By supplementing GF with such a rescaling, methods have been devised to extract quantities such as critical exponents and beta functions in lattice systems [5,6,9]. The method provides an alternative to typical methods like Monte Carlo RG or Dirac eigenmode analyses, and has been put to use in various lattice theories such as 12flavor SU(3) gauge theory and 2-flavor QCD [5,9].…”

Hadronic vacuum polarization using gradient flow

“…While studies of the relation between the RG and the flowtime evolution have also been performed elsewhere in the literature (see, e.g., refs. [26,[91][92][93][94]), to our knowledge the treatment described here has not been discussed before.…”

Stochastic renormalization group and gradient flow

“…An analysis of this equation for discrete, small time steps leads to the stochastic RG instantiation of usual RG scaling laws of correlations of the fundamental field, as well as of composite operators built from it. In particular, by virtue of the stochastic MCRG equivalence, one is led to correlator ratio formulae of the sort described in [3,4], implying a method for measuring anomalous dimensions of scaling operators close to a critical fixed point.…”