Further remarks on $ππ$ scattering dispersion relations

Abstract

The naive use of higher order perturbation theory leads the left--hand cut integrals in $ππ$ 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 $σ$ pole position is determined using the improved method, $M_σ= 483\pm 13 {\rm MeV}$,$Γ_σ= 705 \pm 50{\rm MeV}$. The scattering length parameter is found to be in excellent agreement with the experimental result.