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

We study the zero-dimensional prototype of the path integrals in quantum mechanics and quantum field theory, with the action $S(\phi)=\frac{\sigma }{2}\phi^{2}+\frac{\lambda}{4}\phi^{4}$. 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 $\sigma<0$, 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.