A limit for large $R$-charge correlators in $\mathcal{N}=2$ theories

Using supersymmetric localization, we study the sector of chiral primary operators $(\mathrm{T}\mathrm{r}{\varphi }^2{)}^n$ with large $R$-charge $4n$ in $\mathcal{N}=2$ four-dimensional superconformal theories in the weak coupling regime $g\to 0$, where $\lambda \equiv g^{2}n$ is kept fixed as $n\to \mathrm{\infty }$, $g$ representing the gauge theory coupling(s). In this limit, correlation functions $G_{2n}$ of these operators behave in a simple way, with an asymptotic behavior of the form $G_{2n}\approx F_{\infty}(\lambda) \left(\frac{\lambda}{2\pi e}\right)^{2n}\ n^\alpha $,modulo$O(1/n)$corrections,with$\alpha=\frac{1}{2} $\mathrm{d}\mathrm{i}\mathrm{m}$(\mathfrak{g})$ for a gauge algebra $\mathfrak{g}$ and a universal function $F_{\mathrm{\infty }}(\lambda )$. As a by-product we find several new formulas both for the partition function as well as for perturbative correlators in $\mathcal{N}=2$ $\mathfrak{s}\mathfrak{u}(N)$ gauge theory with $2N$ fundamental hypermultiplets.