Ghost in Scalar State

Abstract

An analysis is made of $s$-wave $\ensuremath{\pi}\ensuremath{\pi}$ scattering under the assumption of the existence of an $\mathrm{SU}(3)$-symmetric scalar nonet satisfying the inverted equal-spacing rule. One finds that in the case of mixing this leads to a result that there is a ghost at negative squared mass, which couples strongly to pseudoscalars. The various implications of this result are analyzed. Specifically, one concludes that the $\ensuremath{\pi}\ensuremath{\pi}$ scattering length is negative and that the forward-backward symmetry in ${\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ production must change sign between 400 and 600 MeV. Scattering lengths in different isospin states are related by using a vector exchange model.