a compilation is provided in [@Lutz:2001yb; @Semke:2011ez] for the first time. The $1/N_c$ expansion is implied by a proper selection of spin-flavour operators in the truncation (\[1-axial-opt\]) (see [@tHooft74; @Witten1979; @Luty1994; @Dai1995] for more technical details). We recall the relevant terms in the chiral Lagrangian $$\begin{aligned}
&& \mathcal{L}^{(1)}= F\, \mathrm{tr} \Big\{ \bar{B}\, \gamma^\mu \gamma_5\, [i\,U_\mu,B]\, \Big\} + D\, \mathrm{tr}\Big\{ \bar{B}\, \gamma^\mu \gamma_5\, \{i\,U_\mu,\,B\}\, \Big\}
\nonumber \\
&& \qquad +\, C\left( \mathrm{tr} \Big\{ (\bar{B}_\mu \cdot i\, U^\mu)\, B\Big\} + \mathrm{h.c.} \right)
+ H\, \mathrm{tr} \Big\{ (\bar{B}^\mu\cdot \gamma_5\,\gamma_\nu   B_\mu)\, i\,U^\nu \Big\}\,,
\nonumber\\
&& U_\mu = {\textstyle{1\over 2}}\,u^\dagger \, \big(
\partial_\mu \,e^{i\,\frac{\Phi}{f}} \big)\, u^\dagger
-{\textstyle{i\over 2}}\,u^\dagger \,(v_\mu+ a_\mu)\, u
+{\textstyle{i\over 2}}\,u \,(v_\mu-a_\mu)\, u^\dagger\;, 
\label{def-FDCH}\end{aligned}$$ where the object $(\bar B_\mu \cdot U^\mu)$ transforms as a flavour octet by construction. For the particular case (\[1-axial-opt\]) the well known hierarchy of sum rules $$\begin{aligned}
&& F= \frac{2}{3}\, D \,,\qquad C=2\,D \,,\qquad H = 3\,D\,, \qquad {\rm for } \qquad \hat g_2 =0  \,,
\nonumber\\
&&  C=2\,D\,,\qquad H= 9\,F-3\,D \,,\qquad\qquad \qquad \! {\rm for } \qquad \hat g_2 \neq 0  \,,
\label{res-FDCHs}\end{aligned}$$ is recovered. At leading order $( N_c)$ there are three sum rules and at subleading order $(N_c^0)$ there remain two sum rules only.

Matrix elements 