parameters. For the former (“fix-parameter”), parameters are the same for all $N_{\max}$, and are chosen to be the fitted values at $N_{\max}=32$. For the latter (“refit-parameter”), parameters are refitted for each $N_{\max}$. The *right panel* shows the hyperfine splittings $M_{\psi(nS)}-M_{\eta_c(nS)}$ as a function of $N_{\max}^{-1}$ ($N_{\max}=L_{\max}$, $m_j=0$) with fixed parameters. The PDG values are marked as crosses. In both figures, different fitting functions, $a+b/N_{\max}+c/N_{\max}^2$ (solid), $a+b \exp(-c N_{\max})$ (dashed), $a+b \exp(-c \sqrt{N_{\max}})$ (dot-dashed), are shown for the fix-parameter results. The refit-parameter results are simply connected by a straight line segments. []{data-label="fig:conv"}](Figures/Hyperfine_splitting "fig:"){width="49.50000%"}

[c ccc ccc ccc]{}

& $\alpha_s(0)$ & $N_f$ & $\mu_\text{g}$ (GeV) & $\kappa$ (GeV) & $m_q$ (GeV) & rms (MeV) & $\overline{\delta_j M}$ (MeV) & $N_\text{exp}$ & $N_{\max}=L_{\max}$\

& & & & 0.985 & 1.570 & 41 & 15 & & 8\
& & & & 0.979 & 1.587 & 32 & 21 & &16\
& & & & 0.972 & 1.596 & 31 & 17 & & 24\
& & & & 0.966 & 1.603 & 31 & 17 & & 32\
& & & & 1.387 & 4.894 & 48 & 6 & & 8\
& & & & 1.392 & 4.899 & 41 & 6 