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

Abstract

We study the zero-dimensional prototype of the path integrals in quantum mechanics and quantum field theory, with the action $S(\ddot{I}•)=\frac{\ddot{I}ƒ}{2}\ddot{I}{•}^2+\frac{\hat{I}}{»}4\ddot{I}{•}^4$. 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 $\ddot{I}ƒ<0$, 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.

1+16 pages, 6 figures, references added, typos corrected