a global fit to the baryon masses from PACS-CS, LHPC, HSC, NPLQCD, QCDSF and ETMC [@PACS-CS2008; @LHPC2008; @HSC2008; @NPLQCD:2011; @Bietenholz:2011qq; @Alexandrou:2013joa]. Based on parameter sets obtained predictions for the baryon sigma terms at physical quark masses and baryon masses for the ensembles of the CLS collaboration are made [@Bruno:2014jqa; @Bruno:2016plf]. Since we do not consider yet lattice discretization effects a systematic error analysis of our results is outside the scope of this work.

Primer on large-$N_c$ sum rules
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Before we turn to the chiral extrapolation of the baryon masses we recall and further develop the large-$N_c$ sum rules previously applied in the studies [@Semke:2011ez; @Semke:2012gs; @Lutz:2014oxa]. They are of crucial importance in a chiral extrapolation study of the baryon masses since they provide a significant parameter reduction at N$^3$LO. The sum rules are a consequence of a large-$N_c$ analysis of baryon matrix elements of axial-vector, vector and pseudo-scalar, scalar quark currents of QCD, which are $$\begin{aligned}
&& A_\mu^{(a)}(x)= \bar q(x) \,\gamma_\mu \,\gamma_5 \,\frac{\lambda^{(a)}}{2} \,q(x)\,, \qquad  V_\mu^{(a)}(x)= \bar q(x) \,\gamma_\mu  \,\frac{\lambda^{(a)}}{2} \,q(x)\,, 
\nonumber\\
&& P^{(a)}(x)= \bar q(x) \, \gamma_5\,\frac{\lambda^{(a)}}{2} \,q(x)\,,\qquad \quad \, S^{(a)}(x)= \bar q(x) \, \frac{\lambda^{(a)}}{2} \,q(x)\,.\end{aligned}$$ On the one hand one may compute such matrix elements from a 