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

Understand

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.