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.