Dispersion Relations for Three-Particle Scattering Amplitudes. I

Abstract

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 ${\mathbf{y}}_{i}={\mathbf{k}}_{i}{(2{m}_{i}E)}^{\ensuremath{-}\frac{1}{2}}$, $\mathbf{y}_{i}^{}{}_{}{}^{\ensuremath{'}}=\mathbf{k}_{i}^{}{}_{}{}^{\ensuremath{'}}{(2{m}_{i}E)}^{\ensuremath{-}\frac{1}{2}}$; $i=1,2,3$, where ${\mathbf{k}}_{1}$, ${\mathbf{k}}_{2}$, ${\mathbf{k}}_{3}$ and ${\mathbf{k}}_{1}$', ${\mathbf{k}}_{2}$', ${\mathbf{k}}_{3}$' are the initial and final momenta of the particles, respectively, and ${m}_{1}$, ${m}_{2}$, ${m}_{3}$ 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 $\frac{N}{D}$ 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.