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

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 $\varphi$ 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 $(\text{Tr} \varphi^2)^n$ admit a finite limit when $g_\text{YM}\to 0$ with large R-charge growing like $\sim 1/g^2_\text{YM}$. The correction with respect to $\mathcal N=4$ correlators is a non-trivial function $F(\lambda; N)$ of the fixed coupling $\lambda=n\,g^2_\text{YM}$ and the gauge algebra rank $N$. We show how to exploit the Toda equation following from the $tt^*$ 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 $(\text{Tr}\varphi^2)^n\,\text{Tr}\varphi^3$ 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.