Left Hand Singularities, Hadron Form Factors and the Properties of the Sigma Meson

By applying analyticity and single channel unitarity we derive a new formula which is useful to analyze the role of the left-hand singularities in hadron form factors and in the determination of the resonance parameters. Chiral perturbation theory is used to estimate the left-hand cut effects in $\pi\pi$ scattering processes. We find that in the $IJ=11$ channel the left-hand cut effect is negligible and in the $IJ=20$ channel the phase shift is dominated by the left-hand cut effect. In the $IJ=00$ channel the left-hand cut contribution to the phase shift has the wrong sign comparing with the experimental data and therefore it necessitates the $\sigma$ resonance. The mass and width of the $\sigma$ resonance is estimated which is found to be sensitive to the experimental value of the scattering length.