{\sqrt{\alpha}}\right) \right\},  
                            \label{bareie}\end{aligned}$$ where ${\rm erfc}\,z = 1 - {\rm erf}\,z$ is the complementary error function. This equation can be solved numerically.

### Ball-Chiu vertex

If any of the minimal Ball-Chiu Ansatz Eq. (\[BCver\]), the Curtis-Pennington Ansatz Eqs.(\[genver\]) and (\[CPTr\]), or the Haeri Ansatz Eq. (\[Hver\]) is used in place of the bare vertex, together with the Landau gauge gluon propagator, we obtain in place of Eq. (\[baresde\]) the equation $$\Sigma(k_4) = \frac{4}{3} \int \frac{d^4k'}{(2\pi)^4} \,
    \frac{\left|{\bf k} - {\bf k}'\right|^2}{(k - k')^2} 
      \frac{\Delta\left((k - k')^2\right)}{ik_4' + \Sigma(k_4')}
         \left[1 + \frac{\Sigma(k_4) - \Sigma(k'_4)}{i(k_4 - k'_4)}\right].  
                                \label{bcsde}$$ It is interesting to note that, within the set of vertex Ansätze we have considered, the heavy quark propagator is insensitive to the transverse part of the vertex. This is not difficult to understand for the Curtis-Pennington vertex, in which the transverse part is heavily damped by the presence of the factor $M^4 \sim m^4$ in the denominator $d(p,q)$. However, in the case of the Haeri vertex there is no such obvious mechanism, and one is led to question whether the heavy quark propagator may be insensitive to a broad class of Ansätze satisfying the criteria specified above Eq. (\[genver\]).

Taking the gluon propagator 