Asymptotically Universal Crossover in Perturbation Theory with a Field Cutoff

TL;DR

This paper investigates the crossover behavior of perturbative coefficients in simple integrals and anharmonic oscillators with a field cutoff, revealing an asymptotic universal form in the large order limit.

Contribution

It introduces an empirical universal approximation for perturbative coefficients in the crossover region, connecting small and large field cutoff regimes.

Findings

Perturbative coefficients follow a Gaussian integral form in the crossover region.

The approach is validated against exact and approximate calculations.

Potential implications for interpolation between renormalization group fixed points.

Abstract

We discuss the crossover between the small and large field cutoff (denoted x_{max}) limits of the perturbative coefficients for a simple integral and the anharmonic oscillator. We show that in the limit where the order k of the perturbative coefficient a_k(x_{max}) becomes large and for x_{max} in the crossover region, a_k(x_{max}) is proportional to the integral from -infinity to x_{max} of e^{-A(x-x_0(k))^2}dx. The constant A and the function x_0(k) are determined empirically and compared with exact (for the integral) and approximate (for the anharmonic oscillator) calculations. We discuss how this approach could be relevant for the question of interpolation between renormalization group fixed points.