Truncating Dyson-Schwinger Equations Based on Lefschetz Thimble Decomposition and Borel Resummation

TL;DR

This work develops a nonperturbative framework for a zero‑dimensional QFT with $S(\phi)=\frac{\sigma}{2}\phi^2+\frac{\lambda}{4}\phi^4$ by combining Lefschetz thimble decomposition with Borel resummation to reconstruct exact correlation functions from divergent perturbative series. It then uses these exact results to design a Dyson–Schwinger truncation scheme based on the large‑$n$ asymptotics of both nonconnected and connected correlators, outperforming traditional truncation, particularly in the $\sigma<0$ (double‑well) regime where nonperturbative saddles matter. The key finding is that perturbative Borel summability alone can be insufficient; including nonperturbative thimble contributions is essential for a complete truncation, and the proposed asymptotics‑driven DS truncation provides a practical path to incorporating resurgence information in DS equations. The results offer a pathway to extending resurgent and thimble methods to more complex QFTs and higher‑dimensional models.

Abstract

We study the zero-dimensional prototype of the path integrals in quantum mechanics and quantum field theory, with the action $S(φ)=\frac{σ}{2}φ^{2}+\fracλ{4}φ^{4}$. Using the Lefschetz thimble decomposition and the saddle point expansion, we derive multiple asymptotic formal series of the correlation function associated with the perturbative and non-perturbative saddle points. Furthermore, we reconstruct the exact correlation function employing the Borel resummation. We then consider how to truncate the Dyson-Schwinger (DS) equations beginning with the perturbation expansion of the correlation functions, analogous to the one obtained from the Feynmann diagram in higher dimensions. For the case $σ<0$, we find that although the asymptotic series around the perturbative saddle point is Borel summable, it does not capture the full information. Consequently, contributions from non-perturbative saddle points must be included to ensure a complete truncation procedure.

Figures (6)

• Figure 1: The upper panels (lower panels) of the figure correspond to $\sigma>0$ ($\sigma<0$), while the left panels (right panels) correspond to ${\rm Im}\lambda>0$ (${\rm Im}\lambda<0$). The solid curves (dashed curves) represent the Lefschetz thimbles (anti-thimbles). The red dots represent the saddle points. Orange curves (blue curves) correspond to the thimbles of $x_0$ ($x_\pm$). The shaded regions represent the regions where the integral converges.
• Figure 2: The $\sigma$ dependence of $r$ for $\lambda=1$.
• Figure 3: In this figure we let $\sigma=1$. The red lines in the figure represent the exact values of $\gamma_2$ calculated using the integral form (\ref{['gamma 2n']}), while the blue dots correspond to the solutions obtained by truncating the DS equations at $\gamma_{2n}$. The upper panels show the traditional truncation with $Q_{2n}(\gamma_2)=0$, and the lower panels show the truncation using \ref{['sigmapasym']}. From the left panels to the right panels, $\lambda=0.1, 1,10$ respectively.
• Figure 4: In this figure $\sigma=-1$, $\lambda=1$. $(a)$ shows the traditional truncation with $Q_{2n}(\gamma_2)=0$. $(b)$ and $(c)$ show the truncation with $Q_{2n}(\gamma_2)={\cal S}[{\cal I}_{2n,0}]$ (\ref{['x0 resum sigma m']}). The blue dots in $(b)$ represents the real parts of the solutions. $(c)$ shows the imaginary parts of the solutions, which approach $0$ as $n$ increases. $(d)$ shows the truncation by replacing $Q_{2n}(\gamma_2)$ by the large $n$ asymptotic behavior of ${\cal I}_{2n}$\ref{['sigmanasym']}.
• Figure 5: $\sigma=1$, $\lambda=1$, $r=0.30737887\cdots$. The left figure shows the truncation with $P(G_2)=0$, and the right figure shows the truncation by using the asymptotic behavior of the connected correlation function (\ref{['G asym']}). A similar result is obtained for different values of $\lambda$.