  Carrying to higher precision the large-$\mathcal{J}$ expansion of Hellerman
and Maeda, we calculate to all orders in $1/\mathcal{J}$ the power-law
corrections to the two-point functions $\mathcal{Y}_n \equiv |x -
y|^{2n\Delta_{\mathcal{O}}} \langle {\mathcal{O}}_n(x) \bar{\mathcal{O}}_n(y)
\rangle$ for generators $\mathcal{O}$ of Coulomb branch chiral rings in
four-dimensional $\mathcal{N} =2$ superconformal field theories. We show these
correlators have the universal large-$n$ expansion \[ \log(\mathcal{Y}_n)
\simeq \mathcal{J} \mathbf{A} + \mathbf{B} + \log(\Gamma( \mathcal{J} + \alpha
+ 1)) , \] where $\mathcal{J} \equiv 2 n \Delta_{\mathcal{O}}$ is the total
$R$-charge of $\mathcal{O}_n$, the $\mathbf{A}$ and $\mathbf{B}$ are
theory-dependent coefficients, $\alpha$ is the coefficient of the Wess-Zumino
term for the Weyl $a$-anomaly, and the $\simeq$ denotes equality up to terms
exponentially small in $\mathcal{J}$. Our methods combine the structure of the
Coulomb-branch effective field theory (EFT) with the supersymmetric recursion
relations. However, our results constrain the power-law corrections to all
orders, even for non-Lagrangian theories to which the recursion relations do
not apply. For the case of $\mathcal{N} = 2$ SQCD, we also comment on the
nature of the exponentially small corrections, which can be calculated to high
precision in the double-scaling limit recently discussed by Bourget et al. We
show the exponentially small correction is consistent with the interpretation
of the EFT breaking down due to the propagation of massive BPS particles over
distances of order of the infrared scale $|x - y|$.
