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

Baryon Chiral Perturbation Theory combined with the $1/N_c$ 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 $1/N_c$ and the low energy power countings are linked according to $1/N_c={\cal{O}}(\xi)={\cal{O}}(p)$. The renormalization to ${\cal{O}}(\xi^3)$ necessary for the mentioned observables is provided, along with applications to the baryon masses and axial couplings as obtained in lattice QCD calculations.