<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>π</mml:mi><mml:mi>π</mml:mi></mml:math>final state interaction in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>K</mml:mi><mml:mo>→</mml:mo><mml:mi>π</mml:mi><mml:mi>π</mml:mi><mml:mo>,</mml:mo><mml:mi> </mml:mi><mml:mi>pp</mml:mi><mml:mo>→</mml:mo><mml:mi>pp</mml:mi><mml:mi>π</mml:mi><mml:mi>π</mml:mi></mml:math>, and related processes

TL;DRAbstract

Final state interactions in the $S$-wave $\ensuremath{\pi}\ensuremath{\pi}$ system $(I=0,2)$ are reexamined on the basis of the Omn\`es-Muskhelishvili equation and the coupled channel formalism. The contributions to the pion scalar form factor from $\ensuremath{\rho}$ 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 $K\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}$ and $\mathrm{pp}\ensuremath{\rightarrow}\mathrm{pp}\ensuremath{\pi}\ensuremath{\pi}$. It is found that the contribution from the ${f}_{0}$ resonance to the form factor is strongly reduced by a nearby zero.

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