New Optimization Methods for Converging Perturbative Series with a Field Cutoff

TL;DR

This paper introduces new optimization techniques for perturbative series in lambda phi^4 problems, leveraging field cutoffs and parameter adjustments to improve convergence and accuracy across different coupling regimes.

Contribution

It proposes novel methods for optimizing perturbative series convergence using field cutoffs and parameter tuning, outperforming existing approaches like LDE at strong and intermediate couplings.

Findings

Optimization at even order is more efficient than at odd order.

The new methods outperform LDE at strong and intermediate coupling.

The approach can be extended to quantum mechanics and quantum field theory.

Abstract

We take advantage of the fact that in lambda phi ^4 problems a large field cutoff phi_max makes perturbative series converge toward values exponentially close to the exact values, to make optimal choices of phi_max. For perturbative series terminated at even order, it is in principle possible to adjust phi_max in order to obtain the exact result. For perturbative series terminated at odd order, the error can only be minimized. It is however possible to introduce a mass shift in order to obtain the exact result. We discuss weak and strong coupling methods to determine the unknown parameters. The numerical calculations in this article have been performed with a simple integral with one variable. We give arguments indicating that the qualitative features observed should extend to quantum mechanics and quantum field theory. We found that optimization at even order is more efficient that at odd order. We compare our methods with the linear delta-expansion (LDE) (combined with the principle of minimal sensitivity) which provides an upper envelope of for the accuracy curves of various Pade and Pade-Borel approximants. Our optimization method performs better than the LDE at strong and intermediate coupling, but not at weak coupling where it appears less robust and subject to further improvements. We also show that it is possible to fix the arbitrary parameter appearing in the LDE using the strong coupling expansion, in order to get accuracies comparable to ours.