Convergence of derivative expansions of the renormalization group

TL;DR

This paper examines the convergence properties of derivative expansions in the exact renormalization group, demonstrating convergence at one and two loops depending on the cutoff type, with implications for higher derivative operators.

Contribution

It provides a detailed analysis of the convergence behavior of derivative expansions in the renormalization group, highlighting the impact of cutoff choices on convergence.

Findings

Converges at one loop with certain smooth cutoffs.

Converges at two loops with smooth exponential cutoff.

Divergent expansions occur with sharp cutoff for higher derivative operators.

Abstract

We investigate the convergence of the derivative expansion of the exact renormalization group, by using it to compute the beta function of scalar field theory. We show that the derivative expansion of the Polchinski flow equation converges at one loop for certain fast falling smooth cutoffs. The derivative expansion of the Legendre flow equation trivially converges at one loop, but also at two loops: slowly with sharp cutoff (as a momentum-scale expansion), and rapidly in the case of a smooth exponential cutoff. Finally, we show that the two loop contributions to certain higher derivative operators (not involved in beta) have divergent momentum-scale expansions for sharp cutoff, but the smooth exponential cutoff gives convergent derivative expansions for all such operators with any number of derivatives.