quantities computed in HQET do not depend on the choice of $m_R$.

After formally carrying out the spatial momentum integration in Eq. (\[hsde2\]), one obtains an equation generically of the form $$\Sigma(K,\kappa) = \frac{1}{\sigma_Q(K,\kappa)} - iK 
  = \int_{-\infty}^\infty dK' \, \left[T(K - K') - T(\kappa - K')\right]
         \sigma_Q(K',\kappa),   \label{frmly}$$ for some kernel $T$, and with $\sigma_Q$ defined by Eq (\[sgdef\]). It is possible to show from this generic form that the effect of making a change of renormalisation point $\kappa_{\rm old} \rightarrow \kappa_{\rm new}$ is equivalent to a shift of the quark propagator solution along the imaginary $K$ axis: $$\sigma_Q(K,\kappa_{\rm new}) = \sigma_Q(K - i\delta m,\kappa_{\rm old}), 
                       \label{shift}$$ where $\delta m$ is the solution to $$\frac{1}{i \kappa_{\rm new}} = 
  \sigma_Q(\kappa_{\rm new} - i\delta m,\kappa_{\rm old}),  \label{delm}$$ that is, the shift is that required to ensure that the new heavy quark propagator pass through the renormalisation point $\sigma_Q(\kappa_{\rm new},\kappa_{\rm new}) = 1/i\kappa_{\rm new}$. Referring to Fig. \[fig1\] we see that this is equivalent to a shift in the position of the origin of the $K$ plane along the $p^2$ axis corresponding to changing $m_R$ by an amount $\delta m$.

One can also demonstrate that the mass differences between any two heavy 