The naive use of higher order perturbation theory leads the left--hand cut integrals in $\pi\pi$ dispersion relations~\cite{ hjy,Xiao01} divergent. This problem is discussed and solved. Also we point out that the Adler zero condition imposes three constraints on the dispersion relations. The $\sigma $ pole position is determined using the improved method, $M_\sigma= 483\pm 13 { \rm MeV}$,$\Gamma_\sigma= 705 \pm 50{ \rm MeV}$. The scattering length parameter is found to be in excellent agreement with the experimental result.
Categories: hep-ph