Precision of chiral-dispersive calculations of π π scattering

Abstract

We calculate the combination 2a⁰_(0)-5a₀(^2) (the Olsson sum rule) and the scattering lengths and effective ranges a₁,a₂^(2) and b₁,b₂^(I) dispersively (with the Froissart-Gribov representation) using, at low energy, the phase shifts for π π scattering obtained by Colangelo, Gasser, and Leutwyler (CGL) from the Roy equations and chiral perturbation theory, plus experiment and Regge behavior at high energy, or directly, using the CGL parameters for a's and b's. We find mismatch, both among the CGL phases themselves and with the results obtained from the pion form factor. This reaches the level of several (2 to 5) standard deviations, and is essentially independent of the detai᾿ls of the intermediate energy region (0.82 ≤ E ≤ 1.42 GeV) and, in some cases, of the high energy behavior assumed. We discuss possible reasons for this mismatch, in particular in connection with an alternate set of phase shifts.