Baryon Chiral Perturbation Theory combined with the $\mathbf{1}\mathbf{/}{\mathbf{N}}_{\mathbf{c}}$ Expansion in SU(3) I: Framework

Baryon Chiral Perturbation Theory combined with the $1\mathrm{/}{N}_c$ expansion is implemented for three flavors. Baryon masses, vector charges and axial vector couplings are studied to one-loop and organized according to the $\xi$-expansion, in which the $1\mathrm{/}{N}_c$ and the low energy power countings are linked according to $1\mathrm{/}{N}_c=\mathcal{O}(\xi )=\mathcal{O}(p)$. The renormalization to $\mathcal{O}({\xi }^3)$ necessary for the mentioned observables is provided, along with applications to the baryon masses and axial couplings as obtained in lattice QCD calculations.