Three-Body Scattering Amplitude. II. Extension to Complex Angular Momentum

We have analyzed the problem of extending the three-body scattering amplitude to complex values of the total angular momentum $J$. We have found four main difficulties: (i) the disconnectedness of the collision matrix; (ii) the complexity of kinematics; (iii) the release of triangular inequalities or of inequalities like $\mathrm{}\left|M\right|<J$, which, when $J$ is complex, transforms finite sums into infinite sums which are most often divergent; and (iv) the presence of complex singularities in cosine angle variables in the full amplitude. This last difficulty is not examined in the present paper. We propose a generalization of the Froissart-Gribov formula for the three-body scattering amplitude. In the nonrelativistic problem, the use of the Fadeev equations takes care of difficulty (i), and difficulty (ii) is smoothed by the use of center-of-mass energies of the three particles and the total angular momentum as the only variables. Of the three natural techniques—using the Schrödinger equation, extending the Fadeev equations, and extending the Fredholm solution of the Fadeev equations to complex values of $J$—only the third one avoids difficulty (iii). We prove that the Fadeev equations cannot be extended because their kernel becomes unbounded and because they do not reduce to the physical equations when $J$ reaches a physical value. However, the Fredholm solution for physical $J$ can be formally extended to complex $J$, and the extended solution is expressed as the quotient of two Fredholm-type series, where each term of the series is analytic in $J$ in a right half-plane. The Sommerfeld-Watson series never converges for the three-particle scattering amplitude, because of difficulties (iii) and (iv).