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

Abstract

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, $S{S}^{+}=1$, 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.

Comments: 10 pages with 4 figures, revised version to appear in Phys. Lett. B