Truncating Dyson-Schwinger Equations Based on Lefschetz Thimble Decomposition and Borel Resummation
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866915093283864576 |
|---|---|
| author | Peng, Feiyu Shu, Hongfei |
| author_facet | Peng, Feiyu Shu, Hongfei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_13364 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Truncating Dyson-Schwinger Equations Based on Lefschetz Thimble Decomposition and Borel Resummation Peng, Feiyu Shu, Hongfei High Energy Physics - Theory Mathematical Physics 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. |
| title | Truncating Dyson-Schwinger Equations Based on Lefschetz Thimble Decomposition and Borel Resummation |
| topic | High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2410.13364 |