Three-body scattering at intermediate energies

Abstract

The Faddeev equation for three-body scattering at arbitrary energies is formulated in momentum space and directly solved in terms of momentum vectors without employing a partial wave decomposition. In its simplest form the Faddeev equation for identical bosons, which we are using, is a three-dimensional integral equation in five variables, magnitudes of relative momenta and angles. This equation is solved through Padé summation. Based on a Malfliet-Tjon-type potential, the numerical feasibility and stability of the algorithm for solving the Faddeev equation is demonstrated. Special attention is given to the selection of independent variables and the treatment of three-body break-up singularities with a spline based method. The elastic differential cross section, semi-exclusive d(N,N ′ ) cross sections and total cross sections of both elastic and breakup processes in the intermediate energy range up to about 1 GeV are calculated and the convergence of the multiple scattering series is investigated in every case. In general a truncation in the first or second order in the two-body t-matrix is quite insufficient.

Electron and photon scattering on three-nucleon bound states

“…In the future it might be advisable to solve that equation directly in vector variables. First steps in that direction have been already undertaken [105]. We still work using partial wave decomposition and would like to indicate some formal structures.…”

First order relativistic three-body scattering

“…where R w (Λ, p m ) is the (standard) Wigner rotation, 12) and p ′ m = Λp m . Since each of the elementary transformations (2.6,2.7,2.8) are unitary, it follows that (2.11) is unitary.…”

“…This is common practice in bound state calculations of few-nucleon systems based on variational [6] and Green's function Monte Carlo (GFMC) methods [7,8,9,10] and was for the first time applied in momentum space Faddeev calculations for bound states in [11] and for scattering at intermediate energies in Ref. [12].…”

Multi-variate Bateman method for two-body scattering without partial-wave decomposition

“…Bateman method could be an effective way of generating arbitrary off-shell T -matrix elements needed in direct momentum-vector approaches to solve three-particle Faddeev equations without employing partial-wave decomposition as in Refs. [19,20].…”

“…There is a growing awareness that, to solve Faddeev equations in momentum space, direct multi-variable approaches might be more appropriate than the traditional procedure of eliminating angular variables via partial-wave expansions [1,18]. In fact, in the context of nuclear three-body problems, considerable progress has been made towards the solution of Faddeev equations without invoking the angular-momentum decomposition [18][19][20]. Therefore, there is considerable interest in multi-variable methods that could produce the three-dimensional off-shell T-matrix in more economical manner than the Nystrom method.…”