dk_4 \, \frac{1}{ik_4' + \Sigma(k_4')} \times}
                  \nonumber \\
& & \left\{\sqrt{\alpha} \left[\frac{\alpha}{2} - (k_4 - k_4')^2 \right] 
      e^{-(k_4 - k_4')^2/\alpha} + \sqrt{\pi} \left|k_4 - k_4'\right|^3 
   {\rm erfc}\left(\frac{\left|k_4 - k_4'\right|}
                     {\sqrt{\alpha}}\right) \right\}
                  \nonumber \\
& & \times   
     \left[1 + \frac{\Sigma(k_4) - \Sigma(k'_4)}{i(k_4 - k'_4)}\right],  
                    \label{bcie}\end{aligned}$$ which can be solved numerically.

### Numerical results: gaussian gluon propagator

For the purpose of determining the analytic structure of the heavy fermion propagator obtained from the DSE with the gaussian gluon propagator Eq. (\[gauss\]), it is sufficient to look at the one parameter family of models obtained by scaling either $\mu$ or $\alpha$ to unity. We choose to scale $\mu$ to unity, which amounts to working with a set of dimensionless quantities $$\hat{k}_4 = k_4/\mu, \hspace{5 mm} \hat{\alpha} = \alpha/\mu^2, 
  \hspace{5 mm} \hat{\sigma}_{\rm Q} = \mu\sigma_{\rm Q}.$$ This choice enables us to recover the infrared dominant model in the limit $\alpha \rightarrow 0$.

In Figs. \[fig2\] and \[fig3\] we plot the heavy quark self energy $\Sigma(k_4)$ as a function of real $k_4$ obtained from the bare vertex DSE Eq. (\[bareie\]) and the Ball-Chiu vertex DSE Eq. (\[bcie\]) for $\hat{\alpha} = 1$, 2 and 3. These results are obtained by iterating from an initial guess and 