\bar g^{(V)}_1 + 3\, \bar g^{(V)}_D - \frac{3}{2}\, \bar h^{(V)}_1 - \frac{3}{2}\, \bar h^{(V)}_3\,,
\label{Q4-subleading}\end{aligned}$$ with $\bar g = g$ and $\bar h = h$. The leading order operators of [@LutzSemke2010] are recovered with $3\,\hat g_1 = \hat g_2$, $3\,\hat g_7 = \hat g_8$ and $\hat g_4 = \hat g_5 = 0 =
\hat g_{10} = \hat g_{11} =0 $. This leads to an additional six leading order sum rules $$\begin{aligned}
&&  \bar h^{(S)}_1 = 0\,,\qquad 
 \bar h^{(S)}_5 =  \bar g^{(S)}_D + 3\, \bar g^{(S)}_1\,\qquad   \bar h^{(S)}_6 = -3\,\Big( \bar g^{(S)}_D + \frac{3}{2}\,\bar g^{(S)}_1 \Big)\,,  
\nonumber\\
&& \bar  h^{(V)}_1= 0 \,, \qquad  \bar g^{(V)}_D = -\frac{3}{2}\,\bar g^{(V)}_1\,,\qquad \quad     \bar h^{(V)}_3 = \frac{3}{2}\, \bar g^{(V)}_1\,.
\label{Q4-leading}\end{aligned}$$ Combining the sum rules of (\[Q4-subleading\], \[Q4-leading\]) confirms the leading order sum rules derived first in [@LutzSemke2010] [^1].

As a further cross check we computed the contributions from the s- and u-channel baryon exchange diagrams. They may modify the sum rules obtained previously and lead to a renormalization of the low-energy constants with $g \to \bar g$ and $h \to \bar h$.

Here we need to be more specific about the kinematics of the correlation function. We assume on-shell baryons with $p_0= \sqrt{M^2+\vec p\,^2} = 