On the exponentially small corrections to $\mathcal{N}=2$ superconformal correlators at large R-charge

Abstract

In this note we consider Coulomb-branch chiral primary correlation functions in $\mathcal{N}=2$ superconformal QCD with gauge group $SU(2)$, in the limit of large R-charge $\mathcal{J}=2n$ for the chiral primary operators $[\mathcal{O}(x){]}^n$ with the inverse gauge coupling $\tau$ held fixed. In previous work, these correlation functions were determined to all orders in $n$, up to unknown exponentially small corrections. In this paper we determine the first several orders of the asymptotic expansion of the exponentially small correction itself. To do this we use: the physical interpretation of the exponentially small correction as the virtual propagation of a massive BPS particle, to fix the leading term in the expansion; the supersymmetric recursion relations to derive differential equations for the coupling-dependence of the subleading terms; and the double-scaling limit, to fix undetermined coefficients in the solution of the differential equation. We calculate the expansion of the exponentially small term up to and including relative order $n^{-\frac{5}{2}}$. We also use the recursion relations to calculate the subleading large-$\mathcal{J}$ corrections to the exponentially small correction in the double-scaling limit, up to and including relative order $n^{-5}$ at fixed double-scaled coupling $\lambda$. We compare the expansion to exact results from supersymmetric localization at the coupling $\tau =\frac{25}{\pi }i$, up to $n=150$. At values $n\sim 100-150$, we find the fixed-coupling and double-scaled large-R-charge expansions are accurate to within one part in $10^6$ and $10^8$, respectively, of the size of the exponentially small correction itself. Relative to the full correlator including the dominant EFT contribution, these estimates give results accuracte to one part in $10^{15}$ and $10^{17}$, respectively.

arXiv comment87 pages, 4 figures, 2 photographs

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