heavy quark DSE of ref. [@BL97].

In summary, our main finding is as follows: Improving the vertex Ansatz does little to improve the analytic structure of the quark propagator. However, improving the Ansatz employed for the gluon propagator, particularly by including a realistic asymptotic ultraviolet tail, moves poles in the heavy quark propagator to a less intrusive part of the complex momentum plane. This portends well for future application of the heavy quark DSE technique.

The layout of the paper is as follows. In Section II we summarise the renormalised quark DSE and the approximations we shall be employing for the quark-gluon vertex and gluon propagator. In Section III we summarise the heavy quark formalism and derive a renormalised DSE for the heavy quark propagator. Numerical solutions to these equations are discussed in Section IV. Conclusions are drawn and suggestions for the direction of future work are given in Section V.

The quark Dyson-Schwinger equation
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Our starting point is the renormalised quark DSE [@RW94] $$\Sigma'(p,\Lambda) = Z_1(\mu^2,\Lambda^2)\frac{4g^2}{3} 
 \int^\Lambda \frac{d^4q}{(2\pi)^4}\, 
D_{\mu\nu}(p - q) \gamma_\mu S(q) \Gamma_\nu(q,p),   \label{DSEqn}$$ where we have used a Euclidean metric in which timelike vectors satisfy $p^2 = -p_{\rm Minkowski}^2 < 0$, and for which $\{\gamma_\mu,\gamma_\nu\} = 