New optimization methods for converging perturbative series with a field cutoff

Abstract

We take advantage of the fact that, in $\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}$ problems, a large field cutoff ${\ensuremath{\varphi}}_{\mathrm{max}}$ makes a perturbative series converge toward values exponentially close to the exact values to make optimal choices of ${\ensuremath{\varphi}}_{\mathrm{max}}.$ For a perturbative series terminated at even order, it is in principle possible to adjust ${\ensuremath{\varphi}}_{\mathrm{max}}$ in order to obtain the exact result. For a perturbative series terminated at odd order, the error can only be minimized. It is, however, possible to introduce a mass shift ${m}^{2}\ensuremath{\rightarrow}{m}^{2}(1+\ensuremath{\eta})$ in order to obtain the exact result. We discuss weak and strong coupling methods to determine ${\ensuremath{\varphi}}_{\mathrm{max}}$ and $\ensuremath{\eta}.$ The numerical calculations in this article are 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 find that optimization at even order is more efficient than optimization at odd order. We compare our methods with the linear $\ensuremath{\delta}$ expansion (LDE) (combined with the principle of minimal sensitivity), which provides an upper envelope for the accuracy curves of various Pad\'e and Pad\'e-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.