QCD Inequalities, Large $N_C$ and $\pi \pi$ Scattering Lengths

In this short note we show that (I) in a QCD-like theory with four (rather than two) degenerate flavors $ud{u}^{\mathrm{\prime }}{d}^{\mathrm{\prime }}$, the $\pi {\pi }^{\mathrm{\prime }}$ scattering length is positive (attractive); and (II) in QCD with only two (u,d) degenerate flavors the I=2 (say, ${\pi }^{+}{\pi }^{+}$ hadronic) scattering length is, in the large $N_C$ limit, repulsive. $\pi ({\pi }^{\mathrm{\prime }})$ are the lowest physical states coupling to $J^p = \bar{u}(x)\gamma_5d(x)$and$J^{p’} = \bar{u}‘(x)\gamma_5d’(x),$ respectively.