Consistency of the Baryon-Multimeson Amplitudes for Large- Nc QCD Feynman Diagrams

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