=   g_D^{(V)}  + g^{(V)}_C\,,\qquad \quad 
\bar g_F^{(V)} =  g_F^{(V)} - g^{(V)}_C\,,\qquad g^{(V)}_C = \frac{C^2}{3\,M^2}\,\alpha_{6}  \,,
\nonumber\\
&&\bar  h_1^{(V)} =  h_1^{(V)}   \,, \qquad \qquad \qquad 
 \bar h_2^{(V)} =  h_2^{(V)}  - \frac{2}{9}\,\frac{H^2}{(M+ \Delta)^2}\,, 
\nonumber \\
&& \bar h_3^{(V)} = h_3^{(V)}  + \frac{4}{27}\,\frac{H^2}{(M+ \Delta)^2} -  \frac{1}{6}\,\frac{\beta_{6}\,C^2}{M\,(M+\Delta)}\,,
\nonumber\\ \nonumber\\
&& \bar h_n^{(S)} = h_n^{(S)} \quad {\rm for} \qquad n \neq 5 \qquad {\rm and }\qquad \bar h_5^{(S)} = h_5^{(S)} +  \frac{C^2}{3\,M}\,\beta_5\,,
\label{Q4-renormalization}\end{aligned}$$ where the coefficients $\alpha_{5,6}$ and $\beta_{5,6}$ depend on the ratio $\Delta/M$ only and approach $ 1$ in the limit with $\Delta \to 0$ (see Appendix A and B).

An analysis of the singular terms may be used to correlate the axial-vector coupling constants, $F,D,C, H$ of the baryon octet and decuplet states. It is more convenient, however, to derive such sum rules in terms of matrix elements of a single axial-vector current (\[1-axial-opt\]).

Chiral extrapolation of baryon masses
=====================================

We consider the chiral extrapolation of the baryon masses to unphysical quark masses. Assuming exact isospin symmetry, the hadron masses are functions of $m_u=m_d\equiv m$ and $m_s$. The ultimate goal is to establish a decomposition of the baryon masses into their power-counting moments $$\begin{aligned}
M_B = \sum_{n=0}^\infty M^{(n)}_B \,,\end{aligned}$$ where there is a 