operators only. Note that the two sums over $c$ in (\[SS-expanded\]) are from $c=0$ to $c=8$. At leading order with $\hat c_6 =\hat c_7 =\hat c_8 =0$ a matching with the tree-level expression derived from the chiral Lagrangian leads to the following 7 sum rules $$\begin{aligned}
&& 2\,c_2=-3\,c_1\,,\qquad 2\,c_0 = c_1 +2\,(c_3+e_0)\,, \qquad 3\,c_1 =e_1\,,\qquad c_4=c_1\,,
\nonumber\\
&& 3\,e_1+2\,e_2 =6\,c_3 \,, \qquad e_3=3\,(c_4+c_5)\,, \qquad c_6 = c_5+e_4 \,.
\label{res-ces}\end{aligned}$$ At subleading order there remain 4 sum rules only $$\begin{aligned}
&& c_0 = 2\,c_3 + e_0 - {\textstyle{ 1\over 6}}\,e_1 - {\textstyle{ 1\over 3}}\,e_2 - {\textstyle{ 1\over 2}}\,c_1\,, \qquad 
c_1 = {\textstyle{ 1\over 3}}\,(e_1 + e_2) - c_2 - c_3\,,
\nonumber\\
&& e_3 = 3\,(c_4 + c_5)\,, \qquad
 e_4 = c_0 + c_2 + c_4 + c_6 - c_3 - c_5 - e_0\,.
\label{ces-subleading}\end{aligned}$$

It is interesting to observe that sum rules (\[res-ces\]) differ from the sum rules applied previously in [@Semke:2011ez; @Semke:2012gs; @Lutz:2014oxa] $$\begin{aligned}
&&c_1 = 2\,c_0, \qquad c_2=-3\,c_0,\qquad  c_3=0 \,, \quad
\nonumber\\
&& e_0 = 0\,, \quad e_1 = 6\, c_0, \qquad  e_2 = -9\,  c_0\,, \qquad  e_3 = 3\, ( c_4 + c_5)\,.
\label{result:large-Nc-chi}\end{aligned}$$ The latter were derived in application of the $\varepsilon$ expansion (\[old-ces\]), where the contribution 