some way towards incorporating some intermediate energy structure by using the computationally convenient gaussian model $$\Delta_{\rm G}(k^2) = \frac{3}{16}(2\pi)^4 \frac{\mu^2}{\alpha^2 \pi^2}
                  e^{-k^2/\alpha}. \label{gauss}$$ We note that the infrared dominant gluon model $\Delta_{\rm IR}$, together with the minimal Ball-Chiu vertex $\Gamma_{\mu \nu}(p,q) = \Gamma_{\mu \nu}^{\rm BC}(p,q)$ defines precisely the light quark model considered in ref. [@BRW92], whereas the gaussian model $\Delta_{\rm G}$ has been used in our earlier rainbow approximation studies of heavy quarks [@BL97]. Below we shall explore the effect on the analytic structure of the heavy quark propagator of combining either $\Delta_{\rm IR}$ or $\Delta_{\rm G}$ with the Ball-Chiu vertex.

A more realistic model gluon propagator which goes some way toward modelling the asymptotically free ultraviolet behaviour of QCD (neglecting logarithmic corrections) has been proposed by Frank and Roberts [@FR96]. It takes the form $$\Delta_{\rm FR}(k^2) = 4\pi^2 d \left[4\pi^2m_t^2\delta^4(k) + 
         \frac{1 - e^{k^2/(4m_t^2)}}{k^2} \right], \label{FRprop}$$ where $d = 12/(33 - 2N_f)$, $N_f = 3$ is the number of light quark flavours, and $m_t \approx 0.69$ GeV is a parameter fitted to a range of calculated pion observables. In our numerical calculations, we find it more convenient to consider a gaussian smeared version of the FR propagator given by 