$λφ^4$ Theory I: The Symmetric Phase Beyond NNNNNNNNLO

TL;DR

This work demonstrates that Schwinger functions in a broad class of Euclidean scalar field theories with $d<4$ are Borel reconstructible via Lefschetz thimbles. Focusing on the 2d $\phi^4$ theory in the $Z_2$ symmetric phase, the authors compute perturbative coefficients up to ${\cal O}(g^8)$ for the vacuum energy and the mass and then apply conformal-mapping and Padé-Borel resummation to access strong-coupling physics. They determine the critical coupling $g_c$ and critical exponents $\nu$ and $\eta$, obtaining $g_c=2.807(34)$, $\nu=0.96(6)$, $\eta=0.244(28)$, and a two-point normalization $\kappa=0.29(2)$, with results in good agreement with lattice and Hamiltonian-truncation methods and consistency with Ising universality. The study highlights a robust, generalizable framework for extracting nonperturbative information from high-order perturbative data, and it lays groundwork for extending the approach to 3d $\phi^4$ and $O(N)$ models.

Abstract

Perturbation theory of a large class of scalar field theories in $d<4$ can be shown to be Borel resummable using arguments based on Lefschetz thimbles. As an example we study in detail the $λφ^4$ theory in two dimensions in the $Z_2$ symmetric phase. We extend the results for the perturbative expansion of several quantities up to N$^8$LO and show how the behavior of the theory at strong coupling can be recovered successfully using known resummation techniques. In particular, we compute the vacuum energy and the mass gap for values of the coupling up to the critical point, where the theory becomes gapless and lies in the same universality class of the 2d Ising model. Several properties of the critical point are determined and agree with known exact expressions. The results are in very good agreement (and with comparable precision) with those obtained by other non-perturbative approaches, such as lattice simulations and Hamiltonian truncation methods.

Figures (9)

• Figure 1: The vacuum energy $\Lambda$ (left) and the mass gap $M$ (right) as a function of the coupling constant $g$ obtained by Borel resumming the perturbative series using the coefficients up to the $g^8$ order. The values reported of $\Lambda$ and $M$ for $g>g_c\approx 2.8$, where a phase transition occurs, refer to the extrapolation using the Borel resummed function.
• Figure 2: The divergent one-loop diagram is exactly canceled by the mass counterterm. Within this scheme we can forget all the diagrams with lines that start and end at the same quartic vertex, their contribution being zero.
• Figure 3: Estimate of the error in the resummation of the physical mass at coupling $g=2$. The blue points are the values of $F_{B,k}^{(N=8)}$ as function of the parameters $s$ and $b$. The red dashed line is the central value of the resummation obtained for $s_0=5/4$, $b_0=17/2$. The red band is the final error on the resummation computed as in \ref{['eq:Err_CM']}.
• Figure 4: (Left panel) The vacuum energy $\Lambda$ as a function of the coupling constant $g$ obtained by ordinary perturbation theory up to the $g^8$ order (blue dashed line), optimal truncation (red dotted line) and Borel resummation using conformal mapping (black solid line). Notice how optimal truncation gives accurate predictions up to $g\lesssim 0.4$, in contrast to blind perturbation theory that breaks down for smaller values of the coupling $g\lesssim 0.2$. For $g\gtrsim 0.4$ the optimal truncation estimate breaks down, since we run out of perturbative terms. Errors are not reported to avoid clutter. (Right panel) Central value and error of $\Lambda(g=1)$ as a function of the $g^N$ terms kept in the conformal mapping resummation technique.
• Figure 5: (Left panel) The vacuum energy $\Lambda$ as a function of the coupling constant $g$ using conformal mapping at different orders: $N=5$ (red line), $N=6$ (green line), $N=7$ (blue line), $N=8$ (black line). Errors are not reported to avoid clutter. The $N=7$ and $N=8$ lines are indistinguishable. (Right panel) Comparison between the results obtained using conformal mapping at $N=8$ (light blue), Padé-Borel approximants (light red) and the results of ref.Elias-Miro:2017xxf (black).