Schwinger-Dyson approximants

TL;DR

This work analyzes the Schwinger‑Dyson equations for the zero‑dimensional λφ^4 theory with m^2>0, introducing Schwinger‑Dyson rational approximants that solve the truncated SD system for all even n‑point functions ⟨φ^{2k}⟩ without invoking perturbation theory. The authors show that ⟨φ^2⟩ and, to some extent, ⟨φ^4⟩ align with Padé approximants to the weak‑coupling series, while higher n‑point functions yield approximants that are products of Padé approximants to Stieltjes quotients. Crucially, they prove the convergence of these SD approximants to the exact nonperturbative n‑point functions for all λ>0 as the truncation size grows, providing rigorous justification and a practical nonperturbative tool in a tractable toy model. This work clarifies when Padé approximants are optimal and offers a robust alternative via SD‑based rational approximants that are better suited for nonperturbative structure in higher‑point functions.

Abstract

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.

Figures (2)

• Figure 1: Imaginary parts of $\langle \phi^2 \rangle$ and $\langle \phi^6 \rangle$ in the complex $\lambda$ plane. The sign of $\mathfrak{Im}\, \langle \phi^2\rangle$ is minus the sign of $\mathfrak{Im} \, \lambda$. The plot for $\mathfrak{Im} \langle \phi^4 \rangle$ is qualitatively similar. On the other hand, the sign of $\mathfrak{Im}\, \langle \phi^6\rangle$ changes across the upper half plane. The same is true for $\langle \phi^{2k}\rangle$, $k\geq 3$. This implies that $\langle \phi^6\rangle$ and higher n-point functions are not Stieltjes functions.
• Figure 2: Various Schwinger-Dyson approximants to $\langle \phi^2\rangle$. The exact result is depicted in black. The [N,N] approximants - in red - form a decreasing subsequence, while the [N,N+1] approximants - in blue - form an increasing subsequence. Given that these subsequences are bounded and monotonic, it is immediate that each of them has a limit function. What it is less immediate is that both subsequences converge to the same limit function, and that the limit function is the exact n-point function.