the renormalisation point $({\bf 0},\kappa)$, we arrive at the leading order heavy quark DSE $$\begin{aligned}
\Sigma(K,\kappa) & = & \frac{4}{3} \int^\Lambda \frac{d^4k'}{(2\pi)^4}\, 
     \frac{1}{ik_4' + k'^2/(2m_R) + \Sigma(k'_4,\kappa)} \nonumber \\
 & & \times \left\{ \frac{\left|{\bf k}'\right|^2 
          \Delta\left[(K - k'_4)^2 + \left|{\bf k}'\right|^2 \right]}
 {(K - k'_4)^2 + \left|{\bf k}'\right|^2}  - (K \rightarrow\kappa) \right\}.
                                  \label{hsde2}\end{aligned}$$ We show in the Appendix that, with the smeared FR gluon propagator Ansatz Eq. (\[GFRprop\]), and assuming a hierarchy of scales $$m_t,\;\left|K\right|,\;\left|\kappa\right| \ll m_R \ll \Lambda,  \label{hier}$$ the integral in Eq. (\[hsde2\]) is independent of the ultraviolet cutoff $\Lambda$, and the heavy quark self energy behaves like $$\Sigma(K,\kappa) \sim  2id(\kappa - K) \ln\left(\frac{m_R}{m_t}\right),$$ as $m_R\rightarrow\infty$. With the more severely truncated gaussian gluon propagator Eq. (\[gauss\]), the renormalisation point $\kappa$ can be taken to $-i\infty$ and the ${k'}^2/(2m_R)$ term in the denominator of the integrand ignored with impunity.

Choice of renormalisation point
-------------------------------

At the end of the day, physical quantities must be insensitive to the choice of renormalisation point $\kappa$. In this section we note that the freedom to choose the renormalisation point is equivalent to the notion of a ‘residual mass’ in HQET [@FNL92; @N94], that is, the notion that to zeroth order in $1/m_R$, physical 