quark–light antiquark meson states calculated from the Bethe-Salpeter formalism set out in ref. [@BL97] is independent of the renormalisation point.

Results
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Our main concern in this paper is to compare how the analytic structure of the heavy quark propagator solutions is affected by the approximations employed both for the quark-gluon vertex and the gluon propagator. In an earlier work [@BL97] an attempt was made to study the heavy quark–light antiquark meson spectrum using a combination of rainbow DSE and ladder BSE. It was found that, if a simple gaussian Ansatz is used for the gluon propagator, complex conjugate poles occur in the heavy quark propagator which prevent solution of the meson BSE. Below we systematically explore the movement of the poles as the bare vertex of the rainbow approximation is replaced by a Ball-Chiu vertex, and as the gaussian gluon propagator is replaced by the more realistic Frank and Roberts propagator.

Gaussian gluon propagator
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### Rainbow approximation

As noted in the previous section, if the gluon propagator Eq. (\[gauss\]) is used, we may set the renormalisation point $\kappa = -i\infty$ and ignore the ${k'}^2/(2m_R)$ term in Eq. (\[hsde2\]). Furthermore, there is no need to distinguish between the independent 