momentum variables $K$ and $k_4$. With these simplifications the Landau gauge, rainbow heavy quark DSE becomes $$\Sigma(k_4) = \frac{4}{3} \int \frac{d^4k'}{(2\pi)^4} \,
    \frac{\left|{\bf k} - {\bf k}'\right|^2}{(k - k')^2} 
      \frac{\Delta\left((k - k')^2\right)}
                {ik_4' + \Sigma(k_4')}.  \label{baresde}$$ Choosing the infrared dominant gluon propagator Eq. (\[delpr\]), the DSE reduces to an algebraic equation with solution $$\Sigma(k_4) =
\left \{ \begin{array}{lrl}
\frac{1}{2} \left(-ik_4 + \sqrt{\frac{3\mu^2}{4} - k_4^2} \right)
         & \mbox{if } & 0\le k_4 < \frac{\sqrt{3}\mu}{2} , \\
                                  &            &            \\
\frac{i}{2} \left(- k_4 + \sqrt{k_4^2 - \frac{3\mu^2}{4}} \right)
         & \mbox{if } & k_4\ge \frac{\sqrt{3}\mu}{2} ,
                   \end{array}   \right.   \label{baresoln}$$ or, using the definition (\[sgdef\]), $$\sigma_{\rm Q}(k_4) =
\left \{ \begin{array}{lrl}
\frac{8}{3\mu^2} \left(-ik_4 + \sqrt{\frac{3\mu^2}{4} - k_4^2} \right)
         & \mbox{if } & 0\le k_4 < \frac{\sqrt{3}\mu}{2} , \\
                                  &            &            \\
\frac{8i}{3\mu^2} \left(- k_4 + \sqrt{k_4^2 - \frac{3\mu^2}{4}} \right)
         & \mbox{if } & k_4\ge \frac{\sqrt{3}\mu}{2} ,
                   \end{array}   \right.   \label{delsig}$$

Alternatively, choosing the gaussian gluon propagator Eq. (\[gauss\]) and carrying out the $d^3{\bf k}$ integration, we obtain the integral equation [@GR80] $$\begin{aligned}
\lefteqn{\Sigma(k_4) = \frac{\mu^2}{2\alpha^2\sqrt{\pi}} 
   \int_{-\infty}^\infty 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|}
                     