Coupled channel effects in pion pion S-wave interaction

Abstract

We study coupled channel effects upon isospin I=2 and I=0 $ππ$ S-wave interaction. With introduction of the $ππ\toρρ\toππ$ coupled channel box diagram contribution into $ππ$ amplitude in addition to $ρ$ and $f_2 (1270)$ exchange, we reproduce the $ππ$ I=2 S-wave and D-wave scattering phase shifts and inelasticities up to 2 GeV quite well in a K-matrix formalism. For I=0 case, the same $ππ\toρρ\toππ$ box diagram is found to give the largest contribution for the inelasticity among all possible coupled channels including $ππ\toωω\toππ$, $ππ\to K \bar{K}\toππ$. We also show why the broad $σ$ appears narrower in production processes than in $ππ$ scattering process.