The $ππ$ Final State Interaction in $K\toππ$, $pp\to ppππ$ and Related Processes

Abstract

Final state interactions in the $S$--wave $ππ$ system (I=0,2) are re-examined on the basis of the Omnès-Mus\-khe\-li\-shvili equation and the coupled channel formalism. The contributions to the pion scalar form factor from $ρ$ and $f_2(1270)$ exchange in the $t$--channel and from the $f_0(980)$ $s$--channel resonance are separately evaluated and the role of the nontrivial polynomial in the Omnès function in a coupled channel situation is elucidated. Applications are made to $K\to ππ$ and $pp\to ppππ$. It is found that the contribution from the $f_0$ resonance to the form-factor is strongly reduced by a nearby zero.