THE REACTION PI-N-]PI-PI-N AT THRESHOLD IN CHIRAL PERTURBATION-THEORY

In the framework of heavy baryon chiral perturbation theory, we give t he 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 a gree 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 u ncertainties mostly related to resonance excitation which make an accu rate 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 serie s converges and one finds a(0)(2) = -0.031 +/- 0.007 consistent with t he one-loop chiral perturbation theory prediction.