field of the Goldstone bosons is $\Phi$. Throughout this chapter we apply the convenient flavour ’$\cdot $’ product notation for terms involving the symmetric baryon fields $B_\mu$ and $\bar B_\mu$ (see [@Lutz:2001yb; @Semke:2011ez]). For instance the product of the two flavour decuplet fields $\bar B_\mu \cdot B_\nu $ is constructed to transform as a flavour octet.

Matrix elements of an axial-vector current 
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We recall the well known operator analysis for the axial-vector current, which correlate the axial-vector coupling constants, $F,D,C,H$ of the baryon octet and decuplet states [@Dashen1994; @Dashen1995]. The matrix elements at zero momentum transfer, $$\begin{aligned}
 &&{\langle \bar p\,,\,\bar\chi\,|}\,A^{(a)}_i(0)\,
{|\,p\,,\chi \rangle} = {(\bar\chi\,|}\,  \hat g_1\,G^{a}_i + \hat g_2\,\big\{ J_i\,,\,T^a\big\}\,
{|\chi )} + {\mathcal O}\left(\frac{1}{N_c} \right)\,,
\label{1-axial-opt}\end{aligned}$$ are parameterized in terms of $\hat g_1$ and $\hat g_2$ and the spin, $J_i$, flavour, $T^a$ and spin-flavour, $G_i^a$ operators [@LutzSemke2010]. It suffices to compute matrix elements at $N_c=3$ only for which the effective spin-flavour states ${|\,p\,,\chi \rangle}$ are introduced. As emphasized in [@Lutz:2001yb; @Semke:2011ez; @Semke:2012gs] the hierarchy of the large-$N_c$ sum rules can efficiently be derived once a complete compilation of the action of the three operators $J_i$, $T^a$ and $G_i^a$ on the spin-flavour states ${|\,p\,,\chi \rangle}$ is available. Such 