Schwinger-Dyson approximants
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918191804973056 |
|---|---|
| author | Fiol, Bartomeu Gijon, Elena Alonso, Unai Lejarza |
| author_facet | Fiol, Bartomeu Gijon, Elena Alonso, Unai Lejarza |
| contents | We revisit the solution to the Schwinger-Dyson equations in the simple case of the 0-dimensional $\frac{1}{2}m^2 ϕ^2 +\fracλ{4} ϕ^4$ theory with $m^2>0$ and $λ\geq 0$. We argue that the truncated Schwinger-Dyson equations are solved by rational approximants to all n-point functions $\langle ϕ^{2k} \rangle$, and provide strikingly simple recursive relations for them. These rational approximants are constructed without any reference to ordinary perturbative expansions. They turn out to be Padé approximants for $\langle ϕ^2 \rangle$ and for half of the truncations in the case of $\langle ϕ^4 \rangle$, but they are not Padé approximants for higher n-point functions. This difference is related to the fact that $\langle ϕ^2 \rangle$ and $\langle ϕ^4 \rangle$ are Stieltjes functions, while higher n-point functions are not. We prove that as the size of the truncation tends to infinity, these rational approximants converge to the full non-perturbative n-point functions for all positive values of the coupling $λ$. Thus, in the example studied in this work, these new rational approximants are much easier to derive than the usual Padé approximants, and when different, they are better suited to approximate the full non-perturbative n-point functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_05665 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Schwinger-Dyson approximants Fiol, Bartomeu Gijon, Elena Alonso, Unai Lejarza High Energy Physics - Theory Mathematical Physics We revisit the solution to the Schwinger-Dyson equations in the simple case of the 0-dimensional $\frac{1}{2}m^2 ϕ^2 +\fracλ{4} ϕ^4$ theory with $m^2>0$ and $λ\geq 0$. We argue that the truncated Schwinger-Dyson equations are solved by rational approximants to all n-point functions $\langle ϕ^{2k} \rangle$, and provide strikingly simple recursive relations for them. These rational approximants are constructed without any reference to ordinary perturbative expansions. They turn out to be Padé approximants for $\langle ϕ^2 \rangle$ and for half of the truncations in the case of $\langle ϕ^4 \rangle$, but they are not Padé approximants for higher n-point functions. This difference is related to the fact that $\langle ϕ^2 \rangle$ and $\langle ϕ^4 \rangle$ are Stieltjes functions, while higher n-point functions are not. We prove that as the size of the truncation tends to infinity, these rational approximants converge to the full non-perturbative n-point functions for all positive values of the coupling $λ$. Thus, in the example studied in this work, these new rational approximants are much easier to derive than the usual Padé approximants, and when different, they are better suited to approximate the full non-perturbative n-point functions. |
| title | Schwinger-Dyson approximants |
| topic | High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2511.05665 |