Putative light scalar nonet

Abstract

We investigate the ``family'' relationship of a possible scalar nonet composed of the ${a}_{0}(980),$ the ${f}_{0}(980)$ and the $\ensuremath{\sigma}$ and $\ensuremath{\kappa}$ type states found in recent treatments of $\ensuremath{\pi}\ensuremath{\pi}$ and $\ensuremath{\pi}K$ scattering. We work in the effective Lagrangian framework, starting from terms which yield ``ideal mixing'' according to Okubo's original formulation. It is noted that there is another solution corresponding to dual ideal mixing which agrees with Jaffe's picture of scalars as $\mathrm{qq}\overline{q}\overline{q}$ states rather than $q\overline{q}$ states. At the Lagrangian level there is no difference in the formulation of the two cases (other than the numerical values of the coefficients). In order to agree with experiment, additional mass and coupling terms which break ideal mixing are included. The resulting model turns out to be closer to dual ideal mixing than to conventional ideal mixing; the scalar mixing angle is roughly $\ensuremath{-}17\ifmmode^\circ\else\textdegree\fi{}$ in a convention where dual ideal mixing is $0\ifmmode^\circ\else\textdegree\fi{}.$