The reaction pi N->pi pi N at threshold in chiral perturbation theory

In the framework of heavy baryon chiral perturbation theory, we give the chiral expansion for the pi N --> pi pi N threshold amplitudes D-1 and D-2 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 pi N --> pi pi N and the pi pi S-wave scattering lengths, a(0)(0) and a(0)(2), respectively, to order O(M(pi)(2)). We show that there are uncertainties mostly related to resonance excitation which make an accurate determination of the pi pi scattering length a(0)(0) from the pi pi N threshold amplitudes at present very difficult. The situation is different in the pi pi isospin two final state. Here, the chiral series converges and one finds a(0)(2) = -0.031 +/- 0.007 consistent with the one-loop chiral perturbation theory prediction.