Scalar radius of the pion and two photons into two pions. Strong S-wave final state interactions

Abstract

The quadratic pion scalar radius, <r^2>^\pi_s, plays an important role for present precise determinations of \pi\pi scattering. The solution of the Muskhelishvili-Omn\`es equations for the non-strange null isospin (I) pion scalar form factor determines that <r^2>^\pi_s=(0.61\pm 0.04) fm^2. However, by using an Omn\`es representation of this form factor, Yndur\'ain recently obtains <r^2>^\pi_s=(0.75\pm 0.07) fm^2. A large discrepancy between both values, given the precision, then results. We show that Yndur\'ain's method is indeed compatible with the determinations from the Muskhelishvili-Omn\`es equations once a zero in the scalar form factor for some S-wave I=0 T-matrices is considered. Once this is accounted for, the resulting value is <r^2>^\pi=(0.63\pm 0.05) fm^2. On the other hand, we perform a theoretical study of the reaction \gamma\gamma\to \pi^0\pi^0 based on dispersion relations. The large source of uncertainty for \sqrt{s}\gtrsim 0.5 GeV, due to variations in the phase used in the Omn\`es function above the K\bar{K} threshold, is removed by taking one more subtraction in the dispersion relation. This allows us to make sharper predictions for the cross section so that one could use this reaction to distinguish between different low energy \pi\pi parameterizations, once independent experiments are available. We also study the role played by the \sigma or f_0(600) meson in this reaction and determine its width to two photons.