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 ${\epsilon }^2$ 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 ${\epsilon }^2$ by applying to the 1PI generator an extension of the operator expansion technique developed by Wegner for the Wilson smooth-cutoff renormalization-group generator.