The Reaction Πn → Ππn at Threshold in Chiral Perturbation Theory

In the framework of heavy baryon chiral perturbation theory, we give the chiral expansion for the πN → ππN threshold amplitudes D1 and D2 to quadratic order in the pion mass. The theoretical results agree within one standard deviation with the empirical values. We also derive a relation between the two threshold amplitudes of the reaction πN → ππN and the ππ S–wave scattering lengths, a0 and a0, respectively, to order O(M2 π). We show that there are uncertainties mostly related to resonance excitation which make an accurate determination of the ππ scattering length a0 from the ππN threshold amplitudes at present very difficult. The situation is different in the ππ isospin two final state. Here, the chiral series converges and one finds a0 = −0.031±0.007 consistent with the one–loop chiral perturbation theory prediction.