The ππ S-wave scattering lengths

We match the known chiral perturbation theory representation of the pi pi scattering amplitude to two loops with a phenomenological description that relies on the Roy equations. On this basis, the corrections to Weinberg's 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, a(0)(0) = 0.220 +/- 0.005, a(0)(2) = -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. (C) 2000 Elsevier Science B,V. All rights reserved.