$$\Delta_{\rm GFR}(k^2) = (2\pi)^4 \frac{m_t^2 d}{\alpha^2 \pi^2}
      e^{-k^2/\alpha} + 4\pi^2 d \frac{1 - e^{k^2/(4m_t^2)}}{k^2}.  
                                          \label{GFRprop}$$ This will enable comparisons to be made with the pure gaussian model $\Delta_G$.

In the next section we consider the heavy quark limit of the quark Dyson-Schwinger equation. Our earlier analysis of this limit  [@BL97] employed the heavily damped gluon propagator $\Delta_{\rm G}$, and so was free from ultraviolet divergences. In order to deal with a more realistic model, such as Eq. (\[GFRprop\]), it is necessary to develop a properly renormalised version of the formalism.

The heavy quark limit
=====================

Heavy quark propagator
----------------------

In the absolute limit of heavy renormalised quark masses, $m_R(\mu^2) \rightarrow \infty$, the dressed quark propagator Eq. (\[genprop\]) is dominated by the bare form $S^{-1}_{\rm bare} = i\gamma\cdot p + m_R$. However it is important to isolate from the full inverse propagator finite order self energy corrections to the bare inverse propagator which drive confining and remnant chiral symmetry breaking effects. To this end we write the momentum variables occurring in the DSE Eq. (\[DSEqn\]) as $$p_\mu = im_R(\mu^2) v_\mu + k_\mu, \hspace{5 mm} 
q_\mu = im_R(\mu^2) v_\mu + k'_\mu,                \label{kmudef}$$ where, for convenience, we take $v = ({\bf 0},1)$, so 