The gradient flow in λφ4 theory

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 A 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 Ac64 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.