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The gradient flow in $λϕ^{4}$ theory

A gradient flow equation for $λϕ^{4}$ theory in $D=4$ is formulated. In this scheme the gradient flow equation is written in terms of the renormalized probe variable $Φ(t,x)$ and renormalized parameters $m^{2}$ and $λ$ in a manner analogous to the higher derivative regularization. No extra divergence is induced in the interaction of the probe variable $Φ(t,x)$ and the 4-dimensional dynamical variable $ϕ(x)$ 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 $λϕ^{4}$ 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.