Consistency of the baryon-multimeson amplitudes for large- N_c QCD feynman diagrams

We study the pion-baryon scattering process π + B → (n − 1)π + B in a QCD theory with a large number (N_c) of colors. It is known that this scattering amplitude decreases with N_c like N_c^1-n/2, and that its individual tree diagrams grow like N_c^n/2. The only way these two can be consistent is for n − 1 powers of N_cto be canceled when the Feynman diagrams are summed. We prove this to be true in tree order for any n.