Truncating Dyson-Schwinger equations based on a 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(ϕ)=σ2ϕ2+λ4ϕ4. 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 nonperturbative saddle points. Furthermore, we reconstruct the exact correlation function employing the Borel resummation. We then consider how to truncate the Dyson-Schwinger equations to compute two-point functions beginning with the perturbation expansion of the correlation functions, analogous to the one obtained from the Feynman 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 nonperturbative saddle points must be included to ensure a complete truncation procedure. Published by the American Physical Society 2025