Stochastic Renormalization Group and Gradient Flow in Scalar Field Theory

Abstract

Recently, the connections between gradient flow and renormalization group have been explored analytically and numerically. Gradient flow (when modified by a field rescaling) can be characterized as a continuous blocking transformation. In this work, we draw a connection between gradient flow and functional renormalization group by describing how FRG can be represented by a stochastic process, and how the stochastic observables relate to gradient flow observables. The relation implies correlator scaling formulae that can be applied numerically in lattice simulations. We present preliminary results on anomalous dimensions obtained from such measurements in the context of 3-dimensional lattice φ 4 theory.

Lattice gauge theory for physics beyond the Standard Model

“…This can be used in conjunction with Monte Carlo lattice simulation to implement a continuous Monte Carlo renormalization group (MCRG) method, which can then be used to extract operator anomalous dimensions. Application of this technique to a gauge-fermion system [122] and scalar field theory in lower dimensions [123] have obtained promising results with relatively low statistics.…”

“…Application of the technique to theories with strongly coupled infrared limit will require a better understanding of the extrapolation to the infrared limit. The behavior of mixing between operators of similar scaling dimension, which can appear as a significant systematic effect if not addressed [123], must be better understood; the variational method, which has been fruitful in the understanding of QCD states which mix with several interpolating operators, may be useful here.…”

Higher derivative extension of the functional renormalization group

“…It seems, however, that the coarse graining procedure that gives the FRG equations including more than two functional derivatives is quite different from the one that gives the FRG equation including up to two derivatives. To our knowledge, such higher derivative FRG equations has not been studied systematically so far, although there is a recent interesting proposal for a manifestly gauge-invariant FRG equation that includes higher functional derivatives [16] (for related works, see [17][18][19][20][21][22][23][24][25][26][27]).…”

Gauge covariant neural network for 4 dimensional non-abelian gauge theory

“…Another example is relation to the renormalization group. Relations of the gradient flow and the renormalization group have been discussed (40)(41)(42)(43). Recently, Sonoda et al found that flowed field can be regarded as smoothly regulated field respecting gauge symmetry, and one can define Polchinskitype renormalization group equation with keeping gauge invariance (42,43).…”