Convergence of derivative expansions in scalar field theory

TL;DR

This paper investigates the convergence properties of derivative expansions in scalar field theory's renormalisation group equations, demonstrating convergence at one and two loops under specific cutoff conditions.

Contribution

It provides the first detailed analysis of convergence of derivative expansions in the exact renormalisation group framework for scalar theories.

Findings

Derivative expansion converges at one loop with certain cutoffs

Legendre flow equation's derivative expansion trivially converges at one loop

Convergence also occurs at two loops with exponential cutoff

Abstract

The convergence of the derivative expansion of the exact renormalisation group is investigated via the computation of the beta function of massless scalar lambda phi^4 theory. The derivative expansion of the Polchinski flow equation converges at one loop for certain fast falling smooth cutoffs. Convergence of the derivative expansion of the Legendre flow equation is trivial at one loop, but also can occur at two loops and in particular converges for an exponential cutoff.