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

Abstract

In this short note we show that (I) in a QCD-like theory with four (rather than two) degenerate flavors $ud{u}^{\prime }{d}^{\prime }$, the $\pi {\pi }^{\prime }$ 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. $\pi ({\pi }^{\prime })$ are the lowest physical states coupling to $J^p=\bar{u}(x){\gamma }_{5}d(x)$ and $J^{p^{\prime }}={\bar{u}}^{\prime }(x){\gamma }_5{d}^{\prime }(x),$ respectively.

8 pages, latex with one postscript figure