Summation of higher order effects using the renormalization group equation

The renormalization group (RG) is known to provide information about radiative corrections beyond the order in perturbation theory to which one has calculated explicitly. We first demonstrate the effect of the renormalization scheme used on these higher order effects determined by the RG. Particular attention is paid to the relationship between bare and renormalized quantities. Application of the method of characteristics to the RG equation to determine higher order effects is discussed and used to examine the free energy in thermal field theory, the relationship between the bare and renormalized coupling, and the effective potential in massless scalar electrodynamics.