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 ${\mathbf{y}}_i={\mathbf{k}}_i{\left(2\mathrm{}{m}_{i}E\right)}^{-\frac{1}{2}}$, $\mathbf{y}_i^{}{}_{}{}^{\prime }=\mathbf{k}_i^{}{}_{}{}^{\prime }{\left(2\mathrm{}{m}_{i}E\right)}^{-\frac{1}{2}}$; $i=1,2,3\mathrm{}$, 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 $T\left(E\right)\mathrm{}$ [defined as a real analytic function: $T\left(E\right)=T^{*}\left(E^{*}\right)$] 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 $T\left(E\right)\mathrm{}$ 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.