of a scalar current 
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Less well studied are matrix elements of the scalar current $$\begin{aligned}
&&{\langle \bar p\,,\,\bar\chi\,|}\,S^{(a)}(0)\,
{|\,p\,,\chi \rangle} = {(\bar\chi\,|}\, \delta_{a0}\,\big(\hat b_1\,{\mathbbm 1} + \hat b_2\,J^2 \big) 
\nonumber\\
&& \qquad \qquad +\, \hat b_3\,T^{a} + \hat b_4\,\big\{ J_i\,,\,G_i^a\big\}\,
{|\chi )} + {\mathcal O}\left(\frac{1}{N^2_c} \right)\,,
\label{1-scal-opt}\end{aligned}$$ where the spin-flavour operators are supplemented by their flavour singlet counter parts $$\begin{aligned}
 T^0 = \sqrt{ \frac 1 6}\, {\mathbbm 1} \,, \qquad G_i^0 =\sqrt{ \frac 1 6}\,J_i \,.\end{aligned}$$ We recall the relevant terms in the chiral Lagrangian $$\begin{aligned}
&& \mathcal{L}^{(2)}_\chi = 2\, b_0 \,\mathrm{tr} \left(\bar B \,B\right) \mathrm{tr}\left(\chi_+\right) + 2 \,b_D\,\mathrm{tr}\left(\bar{B}\,\{\chi_+,\,B\}\right) + 2\, b_F\,\mathrm{tr}\left(\bar{B}\,[\chi_+,B]\right) \nonumber \\
&& \qquad  -\, 2\, d_0\, \mathrm{tr}\left(\bar B_\mu \cdot B^\mu\right) \mathrm{tr}(\chi_+) - 2\, d_D\, \mathrm{tr} \left( \left(\bar{B}_\mu \cdot B^\mu\right) \chi_+\right)\,,
\nonumber\\
&& \chi_+ = {\textstyle{1\over 2}}\, \big( u \,\chi_0 \,u + u^\dagger \,\chi_0 \,u^\dagger \big) = 2\,B_0\, s + \cdots \,,
\label{def-b-d}\end{aligned}$$ where we encounter the scalar source field $s$. For the particular case (\[1-scal-opt\]) the hierarchy of sum rules $$\begin{aligned}
&& b_D = 0\,,\qquad  d_0 - b_0 = - \frac13\,d_D\,,\qquad d_D = 3 \,b_F\, \qquad {\rm for} \qquad \hat b_{2,4} = 0 \,,
\nonumber\\
&& b_F + b_D = \frac13\,d_D\,, \qquad  d_0+ \frac13\,d_D- b_0 