The Constraints of Unitary on $\pi\pi$ Scattering Dispersion Relations

A new dispersion relation for the partial wave $\pi\pi$ 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 $\sigma$ resonance and the $f_0(980)$ resonance are estimated based on the theoretical relations we obtained. The central value of the $\sigma$ pole position is $M_\sigma\simeq 410$MeV, $\Gamma_\sigma\simeq 550$MeV, obtained after including the the constraint of the Adler zero condition.