Closed-form irreducible differential formulations of the Wilson renormalization group

We present a detailed derivation of the one-particle--irreducible (1PI) differential renormalization-group generators originally developed by Nicoll and Chang and by Chang, Nicoll, and Young. We illustrate the machinery of the irreducible formulation by calculating to order epsilon2 the characteristic time exponent z for the time-dependent Ginsburg-Landau model in the cases of conserved and nonconserved order parameter. We then calculate both z and eta to order epsilon2 by applying to the 1PI generator an extension of the operator expansion technique developed by Wegner for the Wilson smooth-cutoff renormalization-group generator