\big(\bar{B}\, (i\,\Dslash -M_{[8]})\, [B, \chi_+]\big) \nonumber\\
&& \qquad + \,\zeta_F\, \mathrm{tr} \big(\bar{B}\, (i\,\Dslash -M_{[8]})\, \{B, \chi_+\} \big) 
- \xi^{}_0\, \mathrm{tr} \big(\bar B_\mu\,\cdot  (i\,\Dslash -M_{[10]})\, B^\mu \big)\, \mathrm{tr}(\chi_+) 
\nonumber \\
&& \qquad -\, \xi^{}_D\, \mathrm{tr} \big( \big(\bar B_\mu\,\cdot (i\,\Dslash -M_{[10]})\,B^\mu \big)\, \chi_+ \big) \,.
\label{def-zeta-xi}\end{aligned}$$ We introduce the leading orders operator ansatz for the correlator as $$\begin{aligned}
&&{\langle \bar p, \, \bar \chi |}\, O^{ab}_i\, {|p,\, \chi\rangle} = (\bar p+p)_i\,{(\bar \chi|}\,
+ \delta_{a0}\,\hat \zeta_1\, T^{b}
\nonumber\\
&& \qquad \qquad+\, \hat \zeta_2\,\{T^{a} ,\,T^{b}\} + \hat \zeta_3\,\{J_i,\,\{T^b,\, G^a_i\}\} \,{|\chi)} + \cdots \,,
\label{def-SV-expansion}\end{aligned}$$ which leads to the following two sum rules $$\begin{aligned}
&& 3\,\zeta_F + 3\,\zeta_D= \xi_D \,,\qquad \qquad  
\xi_0+ \frac{1}{3}\,\xi_D=\zeta_0+ 2\,\zeta_D\,.
\label{def-combinations}\end{aligned}$$ If the subleading operator with $\hat \zeta_3$ in (\[def-SV-expansion\]) would be dropped the additional relation $\zeta_D=0$ arises. Such results are in full analogy to the relations obtained for the symmetry breaking $Q^2$ parameters in (\[res-bds\]).

Product of two scalar currents
------------------------------

0.3cm

We turn to matrix elements of the product of two scalar currents. They probe the symmetry breaking counter terms $c_i$ and $e_i$ of the chiral Lagrangian that are proportional to the square of the current quark masses. We recall their particular form $$\begin{aligned}
\mathcal{L}^{(4)}_\chi &=& c_0\, \mathrm{tr}\left(\bar B\, 