Momentum Scale Expansion of Sharp Cutoff Flow Equations

TL;DR

This paper introduces a systematic expansion method for the exact renormalization group equations with a sharp cutoff, enabling controlled approximations that preserve key invariances and relate to existing local potential approximations.

Contribution

It develops a momentum scale expansion technique for sharp cutoff flow equations, maintaining reparametrization invariance and connecting to known approximations.

Findings

The $O(p^0)$ approximation matches the local potential approximation.

Higher-order $O(p^M)$ approximations systematically improve the flow equations.

Practical difficulties are identified in extending the expansion beyond the lowest order.

Abstract

We show how the exact renormalization group for the effective action with a sharp momentum cutoff, may be organised by expanding one-particle irreducible parts in terms of homogeneous functions of momenta of integer degree (Taylor expansions not being possible). A systematic series of approximations -- the $O(p^M)$ approximations -- result from discarding from these parts, all terms of higher than the $M^{\rm th}$ degree. These approximations preserve a field reparametrization invariance, ensuring that the field's anomalous dimension is unambiguously determined. The lowest order approximation coincides with the local potential approximation to the Wegner-Houghton equations. We discuss the practical difficulties with extending the approximation beyond $O(p^0)$.