The small-$N$ series in the zero-dimensional $O(N)$ model: constructive expansions and transseries

TL;DR

This work analyzes the zero-dimensional $O(N)$ quartic model by studying the partition function $Z(g,N)$ and the free energy $W(g,N)$ on a Riemann surface for the coupling $g$. It proves Borel summability of $Z(g,N)$ and $W(g,N)$ along all rays in the cut plane, and develops a constructive, small-$N$ expansion framework via the intermediate-field representation and Loop Vertex Expansion. The authors derive transseries for $Z(g,N)$, its coefficients $Z_n(g)$, and for the cumulants $W_n(g)$, showing that $Z_n(g)$ contains one-instanton contributions while $W_n(g)$ includes multi-instanton sectors up to $p=n$, with a full transseries for $W(g,N)$ obtained through Möbius inversion. They also establish a tower of differential equations governing these objects, linking perturbative graphs to nonperturbative content and clarifying the interplay between small-$N$ and large-$N$ nonperturbative effects on the Riemann surface. The results provide a rigorous resurgence framework for a zero-dimensional quantum field theory, offering precise analytic control over Stokes phenomena and monodromy across branches of $g$.

Abstract

We consider the 0-dimensional quartic $O(N)$ vector model and present a complete study of the partition function $Z(g,N)$ and its logarithm, the free energy $W(g,N)$, seen as functions of the coupling $g$ on a Riemann surface. Using constructive field theory techniques we prove that both $Z(g,N)$ and $W(g,N)$ are Borel summable functions along all the rays in the cut complex plane $\mathbb{C}_π =\mathbb{C}\setminus \mathbb{R}_-$. We recover the transseries expansion of $Z(g,N)$ using the intermediate field representation. We furthermore study the small-$N$ expansions of $Z(g,N)$ and $ W(g,N)$. For any $g=|g| e^{\imath \varphi}$ on the sector of the Riemann surface with $|\varphi|<3π/2$, the small-$N$ expansion of $Z(g,N)$ has infinite radius of convergence in $N$ while the expansion of $W(g,N)$ has a finite radius of convergence in $N$ for $g$ in a subdomain of the same sector. The Taylor coefficients of these expansions, $Z_n(g)$ and $W_n(g)$, exhibit analytic properties similar to $Z(g,N)$ and $W(g,N)$ and have transseries expansions. The transseries expansion of $Z_n(g)$ is readily accessible: much like $Z(g,N)$, for any $n$, $Z_n(g)$ has a zero- and a one-instanton contribution. The transseries of $W_n(g)$ is obtained using Möebius inversion and summing these transseries yields the transseries expansion of $W(g,N)$. The transseries of $W_n(g)$ and $W(g,N)$ are markedly different: while $W(g,N)$ displays contributions from arbitrarily many multi-instantons, $W_n(g)$ exhibits contributions of only up to $n$-instanton sectors.

Figures (5)

• Figure 1: As $\arg(g)$ increases, the branch cut moves clockwise in the complex $\sigma$-plane. When $g$ crosses the negative real axis the tilted contour is equivalent to a Hankel contour $C$ plus the original contour along the real line \ref{['eq:sigma-contour1']}.
• Figure 2: The cardioid domain $\mathbb{D}_0$ of Eq. \ref{['eq:cardioid']} (dotted blue line) and the extended cardioid $\mathbb{D}_{\theta}$ of Eq. \ref{['eq:W-domain']} (red line), for $\theta=\varphi/6$, in the complex $g$-plane. The branch cut is on the negative real axis, thus the portions of $\mathbb{D}_{\theta}$ going beyond it are to be understood as being on different Riemann sheets.
• Figure 3: Approximate location (see Watson-book) of the Lee-Yang zeros of $Z(g,1)$ (blue dots) in the quadrant $\pi<\varphi<3\pi/2$ of $\mathbb{C}_{3\pi/2}$, together with the boundary of the domain $\mathbb{D}_{\theta}$ (in red).
• Figure 4: Critical points and thimbles (thick lines) in the complex $\phi$-plane. The crosses mark the positions of the instantons and the dashed lines are the tilted contours of Eq. \ref{['DiscontinuityNegativeAxis']}.
• Figure 5: The thimble $\mathcal{J}_{0}$ for $Z_{+}(-|g|)$ (left) and $Z_{-}(-|g|)$ (right) as $|\varphi|\nearrow \pi$.

Theorems & Definitions (27)

• Theorem 1: Nevanlinna-Sokal Sokal:1980ey, extended
• proof
• Proposition 1: Properties of $Z(g,N)$
• proof
• Remark 1
• Proposition 2: Properties of $Z_n(g)$
• proof
• Proposition 3: The LVE expansion, analyticity
• proof
• Remark 2