\bar p_0$. The energy of the current $A^{(a)}_i$ in (\[QCD-identity-AA\]) is $q_0 $. For simplicity we assume the frame where $\vec p+\vec q= 0$, i.e. we have only the two three vectors of $ p$ and $\bar p$ around. In an analysis of the correlation function there are additional terms that are singular in $q_0$, which are a consequence of the s- and u-channel exchange diagrams. The sum rules (\[Q4-subleading\], \[Q4-leading\]) with $\bar g = g$ and $ \bar  h =  h$ follow from the analysis of the contributions that are regular in the limits $q_0 \to 0$ and $q_0 \to \pm\, 2\,M$. However, significant results can only be expected if in addition the poles at $q_0 \to \pm \,2\,M$ are expanded for small $q_0$. This leads to a renormalization of the sum rules. The result of such an analysis is $$\begin{aligned}
&& \bar g_0^{(S)} = g_0^{(S)} - \frac{4}{3}\, g^{(S)}_C \,,  \qquad \quad 
 \bar g_1^{(S)} = g_1^{(S)}  +\frac{1}{3}\,g^{(S)}_C \,,  
\nonumber\\
&& \bar g_D^{(S)} =   g_D^{(S)} + g^{(S)}_C\,,   \qquad \qquad 
 \bar g_F^{(S)} =g_F^{(S)}  - g^{(S)}_C \,,\qquad g^{(S)}_C =  \frac{2\,C^2}{3\,M}\,\alpha_5 \,,
\nonumber\\ 
&& \bar g_0^{(V)} =  g_0^{(V)}  - \frac{4}{3}\, g^{(V)}_C \,, \qquad 
\bar  g_1^{(V)} =  g_1^{(V)}   + \frac{1}{3}\,g^{(V)}_C\,,
\nonumber\\
&& \bar g_D^{(V)} 