Pion-Nucleon Scattering in Baryon Chiral Perturbation Theory combined with the ${ 1/N_c}$ Expansion
This work implements the combined BChPT and 1/Nc expansions for pion-nucleon elastic scattering. The effective theory is based on the baryon sector dynamical spin-flavor SU(4) symmetry emergent in the large Nc limit, whose breaking is controlled by the $1/N_c$ expansion. The non-commutativity of the chiral and 1/Nc expansions in unitarity corrections (loops) requires a linking of both expansions. As it was shown in the case of baryon masses and currents, the natural linking is the $ξ$-expansion, in which $O(p) = O(1/Nc ) = O(ξ)A$. The spin-flavor symmetry requires that the ground state baryons span an SU(4) symmetric irreducible representation which implies that in particular $N$ and $Δ$ are active degrees of freedom in the effective theory. The scattering amplitude is expanded to the next-to-next-to leading order in the $ξ$ expansion, corresponding to the one-loop contributions with the LO Lagrangian. The results are given for generic $N_c$ in order to demonstrate the consistency of the framework. The spin-flavor symmetry plays a central role in maintaining the consistency of the effective theory with respect to the $1/N_c$ expansion. This consistency manifests itself in an improvement in the convergence of the low energy expansion with respect to the case of the ordinary BChPT without an explicit dynamical $Δ$, which is known to be inconsistent with the constraints of $N_c$ scaling. Fits to the $πN \to πN$ S, P and D partial wave amplitudes from the SAID data base are finally used to test the framework and to determine the energy range of its applicability.