Order parameter evolution in scalar QFT: Renormalization group resummation of secular terms

The quantum evolution equations for the field expectation value are analytically solved to cubic order in the field amplitude and to one-loop level in the $\lambda {\phi }^4$ model. We adapt and use the renormalization group (RG) method for such nonlinear and nonlocal equations. The time dependence of the field expectation value is explicitly derived integrating the RG equations. It is shown that the field amplitude for late times approaches a finite limit as ${O(t}^{-3/2})$. This limiting value is expressed as a function of the initial field amplitude.