+ \varepsilon\, c_1^{(0,8)} T^8 
\nonumber\\
&& \qquad \qquad +\, \varepsilon\,c_2^{(0,8)} \{J_i, G^{8}_i\} + \varepsilon^2\, c_2^{(0,27)} \{T^8, T^8\}\, {|\chi)}\,,
\quad 
\label{old-ces}\end{aligned}$$ where for the matching of the symmetry breaking parameters $b_0,b_F, b_D$ and $d_0, d_D$ the first four operators are relevant only. Keeping the first four operators in (\[old-ces\]) the last sum rule in (\[res-bds\]) is obtained.

Product of a scalar and a vector current
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0.3cm

A further set of symmetry breaking low-energy constants of the chiral Lagrangian can be large-$N_c$ correlated upon a study of the current-current correlation with one scalar and one vector current $$\begin{aligned}
&& O^{ab}_\mu (q) = i\,\int d^4 x \,e^{-i\,q\cdot x}  \,{\mathcal T}\,S^{(a)} (x)\,V^{(b)}_\mu(0) \,,
\label{def:scalar-scalar}\end{aligned}$$ in the baryon states. Terms in the chiral Lagrangian that involve a covariant derivative and a scalar source term will contribute to this correlation function. In [@Semke:2012gs] such terms were constructed systematically in terms of the five parameters $\zeta_0,\zeta_D, \zeta_F$ and $\xi_0, \xi_D$, where the flavour structure resembles the one of the $Q^2$ parameters $b_0,b_D, b_F$ and $d_0, d_D$ studied in the previous section. The wave-function renormalization terms of the chiral Lagrangian are recalled with $$\begin{aligned}
&& \mathcal{L}^{(3)}_\chi = \zeta_0\, \mathrm{tr} \big(\bar{B}\, (i\,\Dslash -M_{[8]})\, B\big)\, \mathrm{tr}(\chi_+) + \zeta_D\, \mathrm{tr} 