mass to reproduce the Gell-Mann-Oakes-Renner relation [@Gutsche:2012ez]. Our proposal shares some similarities with other proposals in the literature [@Gutsche:2014oua; @Trawinski:2014msa; @Chabysheva:2012fe].

Running Coupling
----------------

As mentioned, we employ a running coupling based on the 1-loop pQCD. The running coupling is a function of the 4-momentum transfer squared $Q^2=-q^2>0$ (see also Fig. \[fig:runalf\]), viz $$\alpha_s(Q^2) = \frac{1}{\beta_0 \ln \big(Q^2/\Lambda^2 +\tau\big)} \triangleq 
\frac{\alpha_s(M_\textsc{z}^2)}{1+\alpha_s(M_\textsc{z}^2)\beta_0
\ln(\mu^2_\textsc{ir}+Q^2)/(\mu^2_\textsc{ir}+M^2_\textsc{z})},$$ where $\beta_0 = (33-2N_f)/(12\pi)$, with $N_f$ the number of quark flavors, $N_f = 4$ for charmonium and $N_f=5$ for bottomonium. A constant $\tau$ is introduced to avoid the pQCD IR catastrophe. Similar ansätze are widely adopted in the literature (e.g. [@Maris:1999nt]). $\Lambda$ and constant $\tau$ are obtained by fixing the strong coupling at the Z-boson mass $\alpha_s(M_\textsc{z}^2)=0.1183$ and at $Q=0$. In practice, we choose $\alpha_s(0) = 0.6$, corresponding to $\mu_\textsc{ir} = 0.55 \,\mathrm{GeV}$ for $N_f=4$. We find, however, the spectra are not sensitive to the choice of $\alpha_s(0)$ within the range of $0.4\le
\alpha_s(0) \le 0.8$.

![The effective running coupling implemented in this work. Data points correspond to various experimental measurements. The vertical and horizontal lines mark the location of $M_\textsc{z}$ and $\alpha_s(M_\textsc{z}^2)$.[]{data-label="fig:runalf"}](Figures/RunAlf.pdf){width="50.00000%"}

Introducing the evolution of the strong coupling implements asymptotic freedom for the one-gluon exchange through 