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, a⁰₀ and a²₀, respectively, to order (M²_π). We show that there are uncertainties mostly related to resonance excitation which make an accurate determination of the ππ scattering length a⁰₀ 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 a²₀ = -0.031 ± 0.007 consistent with the one-loop chiral perturbation theory prediction.