Dispersion Relations for Three-Particle Scattering Amplitudes. I

We consider the scattering of three nonrelativistic spinless particles interacting via two-body Yukawa potentials. The on-energy-shell T-matrix element is studied as a function of the total center-of-mass kinetic energy E for fixed physical values of the vectors yᵢ=kᵢ(2mᵢE)^-1/2, yᵢ^_^'=kᵢ^_^'(2mᵢE)^-1/2; $i=1,2,3$, where k₁, k₂, k₃ and k₁', k₂', k₃' are the initial and final momenta of the particles, respectively, and m₁, m₂, m₃ are their masses. We show that $T(E)$ [defined as a real analytic function: T(E)=T*(E*)] has no complex singularities in the E plane. Along the real E axis, apart from the expected unitarity branch cuts and the "potential" or left-hand cuts, we find three kinds of anomalous singularities. The first kind arises from the kinematical possibility of the particles undergoing a finite number (depending on the mass ratios) of successive binary collisions ("rescatterings") at arbitrarily large spatial separations. The other two kinds are associated with the existence of two-particle bound states. We show that the discontinuities of $T(E)$ across the anomalous cuts can be explicitly expressed in terms of on-shell physical amplitudes. Accordingly, we formulate N/D equations for the determination of the amplitude. The connection between the rescattering singularities and the convergence of the partial-wave expansion of the amplitude is briefly discussed.