of the counter terms to the baryon masses was considered. Here the fifths term proportional to $\varepsilon^2$ turns relevant. The 12 symmetry breaking counter terms are expressed in terms of the five coefficients in (\[old-ces\]). This leads to the 7 sum rules in (\[result:large-Nc-chi\]).

Product of two axial-vector currents
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There remain the sum rules derived from the product of two axial-vector currents, studied previously in [@LutzSemke2010]. They correlate two-body meson-baryon counter terms. A complete list of chiral symmetry conserving $Q^2$ counter terms was given in [@Lutz2002a; @LutzSemke2010]. Here we recall all terms relevant for the calculation of the N$^3$LO baryon masses. The terms are grouped according to their Dirac structure $ \mathcal{L}^{(2)}=\mathcal{L}^{(S)} + \mathcal{L}^{(V)} $ with $$\begin{aligned}
&& \mathcal{L}^{(S)} =- \frac{1}{2}\,g_0^{(S)}\,\mathrm{tr} \,\Big\{\bar{B}\,B \Big\}\, \mathrm{tr}\Big\{ U_\mu\, U^\mu \Big\} - \frac{1}{2}\,g_1^{(S)}\,\mathrm{tr} \,\Big\{ \bar{B}\, U^\mu \Big\}\, \mathrm{tr}\,\Big\{U_\mu\, B \Big\}
\nonumber \\
&&\qquad -\,\frac{1}{4}\,g_D^{(S)} \mathrm{tr}\,\Big\{\bar{B}\left\{\left\{U_\mu, U^\mu\right\}, B\right\}\Big\}
-\frac{1}{4}\,g_F^{(S)}\mathrm{tr}\,\Big\{ \bar{B}\left[\left\{U_\mu, U^\mu\right\}, B\right]\Big\}
\nonumber\\
&&\qquad + \,\frac{1}{2}\,h_1^{(S)}\,\mathrm{tr}\,\Big\{ \bar{B}_\mu \cdot B^\mu \Big\}\, \mathrm{tr}\,\Big\{U_\nu\; U^\nu\Big\} +
\frac{1}{2}\,h_2^{(S)}\,\mathrm{tr}\,\Big\{\bar{B}_\mu \cdot B^\nu \Big\}\, \mathrm{tr}\,\Big\{U^\mu\, U_\nu\Big\}
\nonumber \\
&& \qquad + \,h_3^{(S)}\,\mathrm{tr}\,\Big\{\Big(\bar{B}_\mu \cdot B^\mu\Big)\, \Big(U^\nu\, U_\nu\Big) \Big\} + \frac{1}{2}\,h_4^{(S)}\,\mathrm{tr}\,\Big\{ \Big(\bar{B}_\mu \cdot B^\nu\Big)\, \{U^\mu,\, U_\nu \} \Big\}
\nonumber \\
&&\qquad  +\, h_5^{(S)}\, \mathrm{tr}\, \Big\{ \Big( \bar{B}_\mu \cdot U_\nu\Big)\, \Big(U^\nu\cdot B^\mu \Big) 