On the large R-charge $\mathcal{N}=2$ chiral correlators and the Toda equation

Abstract

We consider $\mathcal{N}=2$ $SU(N)$ SQCD in four dimensions and a weak-coupling regime with large R-charge recently discussed in arXiv:1803.00580. If $\phi$ denotes the adjoint scalar in the $\mathcal{N}=2$ vector multiplet, it has been shown that the 2-point functions in the sector of chiral primaries $(\text{Tr}{\phi }^2{)}^n$ admit a finite limit when $g_{\text{YM}}\to 0$ with large R-charge growing like $\sim 1/g_{\text{YM}}^2$. The correction with respect to $\mathcal{N}=4$ correlators is a non-trivial function $F(\lambda ;N)$ of the fixed coupling $\lambda =n{g}_{\text{YM}}^2$ and the gauge algebra rank $N$. We show how to exploit the Toda equation following from the $t{t}^{\ast }$ equations in order to control the R-charge dependence. This allows to determine $F(\lambda ;N)$ at order $O({\lambda }^{10})$ for generic $N$, greatly extending previous results and placing on a firmer ground a conjecture proposed for the $SU(2)$ case. We show that a similar Toda equation, discussed in the past, may indeed be used for the additional sector $(\text{Tr}{\phi }^2{)}^n\text{Tr}{\phi }^3$ due to the special mixing properties of these composite operators on the 4-sphere. We discuss the large R-limit in this second case and compute the associated scaling function $F$ at order $O({\lambda }^7)$ and generic $N$. Large $N$ factorization is also illustrated as a check of the computation.

27 pages. v2: minor clarifications added

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