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

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 $\mathbb{C}_{\pi} =\mathbb{C}\setminus \mathbb{R}_-$. 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 $|\varphi|<3\pi/2$, 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 M\"oebius 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.