Comments on some chiral-dispersive calculations of $\pi\pi$ scattering

Abstract

We consider the $\pi\pi$ scattering amplitude obtained by Colangelo, Gasser and Leutwyler (CGL), using dispersion relations and chiral perturbation theory to two loops. We point out that the input used by CGL for energies above 1.42 GeV is incompatible with a number of experimental results. Moreover, the error they adscribe to the $\delta_0^{(0)}$ phase at 0.8 GeV, an important part of their input, is much underestimated, and its central value is probably displaced. Then, we compare CGL results with direct fits to experimental $\pi\pi$ data; the outcome of this test is mismatch of the CGL amplitude, for several quantities by as much as 4 standard deviations. We also evaluate forward dispersion relations with the CGL amplitudes. We find that these dispersion relations are fulfilled less and less well as the energy increases towards 0.8 GeV. Moreover, the size that the experimental results for $\pi\pi$ scattering indicates for the two loop corrections in a chiral perturbation theory analysis is less than those required by CGL, which suggests that at least some of the CGL corrections are so large to cover biases in their $\pi\pi$ amplitude and in the evaluation of the scalar form factor of the pion. We conclude on the necessity of repeating the analysis of CGL with correct high energy amplitudes and a more realistic input for the $\delta_0^{(0)}$ phase at 0.8 GeV.