= 2\,b_D\,, \; \qquad {\rm for} \qquad \hat b_{2} = 0 \,,
\nonumber\\
&& b_F + b_D = \frac13\,d_D\,, \qquad \qquad \qquad \qquad \qquad \qquad \qquad \;{\rm for} \qquad \hat b_{2,4} \neq 0 \,,
\label{res-bds}\end{aligned}$$ is predicted. The relations (\[res-bds\]) are a straight forward consequence of the technical results of [@Dashen1994; @Dashen1995; @LutzSemke2010]. At leading order $( N_c)$ there are three sum rules, at subleading order $(N_c^0)$ there are two sum rules and at the accuracy level $(1/N_c)$ there remains one sum rule only. The leading order symmetry breaking parameters of the chiral Lagrangian $b_0,b_F, b_D$ and $d_0, d_D$ as used in [@Semke:2011ez; @Semke:2012gs; @Lutz:2014oxa] are correlated.

Such relations (\[res-bds\]) were obtained before in [@Jenkins1995], however, with a method slightly distinct to our approach. In [@Jenkins1995] the contribution of the counter terms to the baryon masses was analyzed and expanded in powers of the flavour breaking parameter $$\begin{aligned}
\varepsilon \sim m_s- \frac{1}{2}\,(m_u +m_d)\,,\end{aligned}$$ with the current quark masses $m_{u,d,s}$ of QCD. Accordingly the large-$N_c$ operator expansion was sorted according to powers of the $T_8$ and $G_8^i$ operators. At subleading order in $\varepsilon$, the following operators were considered $$\begin{aligned}
\label{lbl:Nc_baryonmass_epsilon2}
&& M_B = {(\bar \chi|}\, c_1^{(0,1)}\mathbbm 1 + c_2^{(0,1)} J_i\, J_i 