Scalar mesons and low-mass sigma: Does the $\sigma$ reveal the confinement singularity?

Abstract

I present a short review of the current understanding of the scalar meson sector, with special attention to the problem of the low-mass $\sigma$. The dispersion relation $N/D$-method used for the restoration of the low-energy $\pi\pi$ $(IJ^{PC}=00^{++})$-wave amplitude is discussed. The low-energy $\pi\pi$ amplitude was determined from the data in the energy region 280--500 MeV and it was sewn with the previously found $K$-matrix solution for the region 450--1950 MeV. The $N/D$-amplitude has a pole on the second sheet of the complex-$s$ plane, near the $\pi\pi$ threshold at $\sqrt s \simeq 430-i325$ MeV, that corresponds to the low-mass $\sigma$-meson. I discuss the hypothesis that this pole may be related to the confinement forces, thus being the eyewitness of confinement.