Schwinger-Dyson approximants

We revisit the solution to the Schwinger-Dyson equations in the simple case of the 0-dimensional $\frac{1}{2}m^2 \phi^2 +\frac{\lambda}{4} \phi^4$ theory with $m^2>0$ and $\lambda \geq 0$. We argue that the truncated Schwinger-Dyson equations are solved by rational approximants to all n-point functions $\langle \phi^{2k} \rangle$, 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 Pad\'e approximants for $\langle \phi^2 \rangle$ and for half of the truncations in the case of $\langle \phi^4 \rangle$, but they are not Pad\'e approximants for higher n-point functions. This difference is related to the fact that $\langle \phi^2 \rangle$ and $\langle \phi^4 \rangle$ 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 $\lambda$. Thus, in the example studied in this work, these new rational approximants are much easier to derive than the usual Pad\'e approximants, and when different, they are better suited to approximate the full non-perturbative n-point functions.