  We consider $\mathcal N=2$ conformal QCD in four dimensions and the one-point
correlator of a class of chiral primaries with the circular $\frac{1}{2}$-BPS
Maldacena-Wilson loop. We analyze a recently introduced double scaling limit
where the gauge coupling is weak while the R-charge of the chiral primary
$\Phi$ is large. In particular, we consider the case
$\Phi=(\text{tr}\varphi^{2})^{n}$ , where $\varphi$ is the complex scalar in
the vector multiplet. The correlator defines a non-trivial scaling function at
fixed $\kappa = n\,g_{\rm YM}^{2}$ and large $n$ that may be studied by
localization. For any gauge group $SU(N)$ we provide the analytic expression of
the first correction $\sim \zeta(3)\,\kappa^{2}$ and prove its universality. In
the $SU(2)$ and $SU(3)$ theories we compute the scaling functions at order
$\mathcal O(\kappa^{6})$. Remarkably, in the $SU(2)$ case the scaling function
is equal to an analogous quantity describing the chiral 2-point functions
$\langle\Phi\overline\Phi\rangle$ in the same large R-charge limit. We
conjecture that this $SU(2)$ scaling function is computed at all-orders by a
$\mathcal N=4$ SYM expectation value of a matrix model object characterizing
the one-loop contribution to the 4-sphere partition function. The conjecture
provides an explicit series expansion for the scaling function and is checked
at order $\mathcal O(\kappa^{10})$ by showing agreement with the available data
in the sector of chiral 2-point functions.
