The ππ S-wave scattering lengths

We match the known chiral perturbation theory representation of the ππ scattering amplitude to two loops with a phenomenological description that relies on the Roy equations. On this basis, the corrections to Weinbergʹs low energy theorems for the S-wave scattering lengths are worked out to second order in the expansion in powers of the quark masses. The resulting predictions, a00=0.220±0.005, a02=−0.0444±0.0010, contain remarkably small uncertainties and thus allow a very sensitive experimental test of the hypothesis that the quark condensate is the leading order parameter of the spontaneously broken chiral symmetry.