Further remarks on $\pi \pi$ scattering dispersion relations

Abstract

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, $M_{\sigma }=483\pm 13\mathrm{MeV}$,${\Gamma }_{\sigma }=705\pm 50\mathrm{MeV}$. The scattering length parameter is found to be in excellent agreement with the experimental result.

Comments: replace with revised version to appear in Chinese Physics Letters. Numerical calculation improved. $M_{\sigma }\simeq 483\mathrm{MeV}$,${\Gamma }_{\sigma }\simeq 705\mathrm{MeV}$