The ππ\pi\pi Final State Interaction in K→ππK\to\pi\pi, pp→ppππpp\to pp\pi\pi and Related Processes

Final state interactions in the $S$--wave $\pi\pi$ system (I=0,2) are re-examined on the basis of the Omn\`es-Mus\-khe\-li\-shvili equation and the coupled channel formalism. The contributions to the pion scalar form factor from $\rho$ 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 Omn\`es function in a coupled channel situation is elucidated. Applications are made to $K\to \pi\pi$ and $pp\to pp\pi\pi$. It is found that the contribution from the $f_0$ resonance to the form-factor is strongly reduced by a nearby zero.Comment: PACS numbers: 13.75.Lb, 13.25.Es, 13.25.+m, 15 pages in plain latex with epsf.sty, 10 figure

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