# Distribution of solutions of the fastest apparent convergence condition in optimized perturbation theory and its relation to anti-Stokes lines

## TL;DR

This paper analyzes the distribution of solutions to the fastest apparent convergence condition in optimized perturbation theory, revealing how zeros accumulate on anti-Stokes lines, which explains the insensitivity of physical quantities to parameter choices.

## Contribution

It introduces a novel integral representation and applies Lefschetz thimble theory to understand the zero distribution in OPT.

## Key findings

- Zeros of the FAC condition accumulate on anti-Stokes lines as perturbation order increases.
- The accumulation explains the insensitivity of OPT results to artificial parameters.
- Provides a new perspective on the mathematical structure underlying optimized perturbation theory.

## Abstract

We discuss fundamental properties of the fastest apparent convergence (FAC) condition which is used as a variational criterion in optimized perturbation theory (OPT). We examine an integral representation of the FAC condition and a distribution of the zeros of the integral in a complex artificial parameter space on the basis of theory of Lefschetz thimbles. We find that the zeros accumulate on a certain line segment so-called anti-Stokes line in the limit $K \to \infty$, where $K$ is a truncation order of a perturbation series. This phenomenon gives an underlying mechanism that physical quantities calculated by OPT can be insensitive to the choice of the artificial parameter.