+ 2\frac{(k_4 - k'_4)^2}{(k - k')^2}
      \right]   \nonumber \\
   & & \times \Delta\left[(k - k')^2\right] \frac{1}{iK'+\Sigma(K',\kappa)}
                    + O\left(\frac{1}{m_R}\right) , \label{AprSDE}\end{aligned}$$ and $$B'(p^2,\Lambda^2) = 2Z_1  \int^\Lambda \frac{d^4k'}{(2\pi)^4}\,
   \Delta\left[(k - k')^2\right] \frac{1}{iK'+\Sigma(K',\kappa)}
                    + O\left(\frac{1}{m_R}\right) . \label{BprSDE}$$ From Eqs. (\[AApr\]), (\[BBpr\]), (\[sigAdef\]), (\[sigBdef\]) and (\[Sigdef\]) we have $$\Sigma(K,\kappa) = \left[B'(p^2,\Lambda^2) - m_R A'(p^2,\Lambda^2)\right]
                    -[p^2\rightarrow\mu^2],$$ which gives $$\Sigma(K,\kappa) = \frac{4}{3} 
   \int^\Lambda \frac{d^4k'}{(2\pi)^4}\,
 \left[ \frac{\left|{\bf k} - {\bf k}'\right|^2}{(k - k')^2} 
   \Delta\left[(k - k')^2\right] \frac{1}{iK'+\Sigma(K',\kappa)}
           - (K\rightarrow\kappa) \right] , \label{hsde1}$$ where $K$ on the left hand side is related to $k$ under the integrand via Eq. (\[Kdef\]), and a similar definition exists relating $K'$ and $k'$. Here we have assumed that Eqs. (\[Zfix\]) and (\[AprSDE\]) imply $Z_1 = 1 + O(1/m_R)$.

As noted above, $K$ and $k_4$ are interchangeable to leading order, except where they occur in the denominator of the heavy fermion propagator $1/[iK' + \Sigma(K')]$ in the integrand of Eq. (\[hsde1\]). This is because, for any gluon propagator $\Delta$ with a realistic asymptotic UV behaviour, all powers of $k'$ must be retained in the denominator to maintain the same degree of divergence in the heavy fermion DSE as in the original equation (\[DSEqn\]). With this observation, and choosing $k_\mu = ({\bf 0},K)$ and 