QCD Inequalities, Large $N_C$ and $ππ$ Scattering Lengths

Abstract

In this short note we show that (I) in a QCD-like theory with four (rather than two) degenerate flavors $ud u'd'$, the $ππ'$ scattering length is positive (attractive); and (II) in QCD with only two (u,d) degenerate flavors the I=2 (say, $π^+π^+$ hadronic) scattering length is, in the large $N_C$ limit, repulsive. $π(π')$ are the lowest physical states coupling to $J^p = \bar{u}(x)γ_5d(x)$ and $J^{p'} = \bar{u}'(x)γ_5d'(x),$ respectively.