Large-$N_c$ and Chiral Limits of QCD and Models of the Baryon

Two ideas have greatly contributed to our understanding of baryon structure in the framework of Quantum Chromodynamics (QCD). The first, chiral symmetry, received its fundamental justification from QCD and has been developed into the powerful technique of chiral perturbation theory. The other notion is the large-$N_c$ limit, namely the behavior of QCD in the limit where there are a large number of colors. Like chiral symmetry, the large-$N_c$ limit predicts many relations between baryon matrix elements. These so-called ``model-independent'' predictions should agree with each other, but paradoxically that is not always the case. The paradox is resolved by recognizing that the two limiting processes (large-$N_c$ and the chiral limit) do not commute. This noncommutivity can be traced to the special role played by the $Δ$ resonance in large-$N_c$ QCD. This gives rise to a new parameter in the effective theory of baryons.