The Constraints of Unitary on $ππ$ Scattering Dispersion Relations
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A new dispersion relation for the partial wave $ππ$ scattering $S$ matrix is set up. Using the dispersion relation we generalize the single channel unitarity condition, $SS^+=1$, to the entire complex $s$ plane, which is equivalent to the generalized unitarity condition in quantum mechanics. The pole positions of the $σ$ resonance and the $f_0(980)$ resonance are estimated based on the theoretical relations we obtained. The central value of the $σ$ pole position is $M_σ\simeq 410$MeV, $Γ_σ\simeq 550$MeV, obtained after including the the constraint of the Adler zero condition.
10 pages with 4 figures, revised version to appear in Phys. Lett. B
10 pages with 4 figures, revised version to appear in Phys. Lett. B