Theoretical Aspects of the Pion-pion Interaction

The structure of the ππ interaction at low energies is strongly constrained by the fact that the pion is the Goldstone boson of a spontaneously broken approximate symmetry of QCD.1 The scattering amplitude can be calculated in a systematic way in the framework of Chiral Perturbation Theory (ChPT),2 by an expansion which converges very rapidly near the center of the Mandelstam triangle. However, the convergence becomes slow as one approaches the unitarity cuts, and already at threshold the direct application of chiral expansions is not satisfactory. The dispersion relations determine the structure of the amplitude in terms of physical region absorptive parts and two subtraction constants, which can be identified with the S-wave scattering lengths, a0 and a 2 0. By projecting the fixed-t dispersion relations onto partial waves and using unitarity, one obtains a set of integral equations for the phase shifts, the Roy equations.3 In the early applications of Roy equations the subtraction constants were taken from experiment, and had large uncertainties. An important step forward was to determine them theoretically, by combining ChPT inside the convergence region with dispersion relations, which led to a very precise theoretical prediction:4 a00 = 0.22± 0.005 , a20 = −0.0444± 0.0010 . (1)