The gradient flow in $\lambda\phi^{4}$ theory

TL;DR

This paper formulates a gradient flow equation for the four-dimensional      theory using renormalized variables, demonstrating its finiteness and applicability beyond Yang-Mills theories.

Contribution

It introduces a gradient flow formulation for   theory in four dimensions, showing its all-order perturbative finiteness and broad applicability.

Findings

Gradient flow equation formulated in terms of renormalized variables.

Finiteness established to all orders in perturbation theory.

Demonstrates the generality of gradient flow beyond Yang-Mills theories.

Abstract

A gradient flow equation for $\lambda\phi^{4}$ theory in $D=4$ is formulated. In this scheme the gradient flow equation is written in terms of the renormalized probe variable $\Phi(t,x)$ and renormalized parameters $m^{2}$ and $\lambda$ in a manner analogous to the higher derivative regularization. No extra divergence is induced in the interaction of the probe variable $\Phi(t,x)$ and the 4-dimensional dynamical variable $\phi(x)$ which is defined in renormalized perturbation theory. The finiteness to all orders in perturbation theory is established by power counting argument in the context of $D+1$ dimensional field theory. This illustrates that one can formulate the gradient flow for the simple but important $\lambda\phi^{4}$ theory in addition to the well-known Yang-Mills flow, and it shows the generality of the gradient flow for a wider class of field theory.