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-fourth model. We adapt and use the renormalization group (RG) method for such non-linear and non-local 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 the time to the power -3/2. This limiting value is expressed as a function of the initial field amplitude.