We study the Nc scalings of pion-nucleon and nucleon-nucleon scatterings in hadron effective field theory. By assuming Witten's counting rules are applied to matrix elements or scattering amplitudes which use the relativistic normalization for the nucleons, we find that the nucleon axial coupling gA is of order N 0 c , and a consistent large Nc counting can be established for the pionnucleon and nucleon-nucleon scatterings. We also justify the nonperturbative treatment of the low energy nucleon-nucleon interaction with the large Nc analysis and find that the deuteron binding energy is of order 1/Nc.
PACS numbers:In the seminal paper Ref.[1], 't Hooft showed that QCD has a hidden expansion parameter 1/N c . With the 1/N c expansion the QCD coupling constant g is of order 1/ √ N c , and gluons are denoted by double lines. By counting the number of interaction vertexes and closed color loops, one can figure out the N c order of a specific Feynman diagram. It is then found that in the large N c limit, the leading contribution comes from planar diagrams with minimal quark loops. The idea to expand QCD in 1/N c is very attractive, as it explains hadron phenomenology successfully, for instance the OZI suppression rule. The extension of the 1/N c expansion to baryons was first carried out by Witten [2]. The 1/N c expansion of baryons, which are bound states of N c valence quarks and have masses of order N c , is more complicated than that of mesons. Using quarks and gluons as degrees of freedom, Witten showed that meson-baryon coupling is of order √ N c , meson-baryon scattering amplitude is of order N 0 c , and baryon-baryon scattering amplitude is of order N c (these results are called the large N c counting rules of Witten in this manuscript). It is interesting to study whether a realistic hadron effective field theory using baryons and mesons as degrees of freedom can reproduce the large N c counting rules of Witten. We will see in the following that this is not a clearly solved problem.