Abstract
To extend the applications of the so-called ``three-dimensional'' formalism to the description of three-nucleon scattering within the Faddeev formalism, we develop a general form of the three-nucleon scattering amplitude. This form significantly decreases the numerical complexity of the ``three-dimensional'' calculations by reducing the scattering amplitude to a linear combination of momentum-dependent spin operators and scalar functions of momenta. The number and structure of the spin operators is fixed and the scalar functions can be represented numerically using standard methods such as multidimensional arrays. In this paper, we show that all orders of the iterated Faddeev equation can be written in this general form. We argue that calculations utilizing the three-nucleon force will also conform to the same general form. Additionally, we show how the general form of the scattering amplitude can be used to transform the Faddeev equation to make it suitable for numerical calculations using iterative methods.
Highlights
A general operator form of the three-nucleon (3N ) scattering amplitude has potential applications in calculations that employ the so-called three dimensional (3D) formalism to calculate observables in the nucleon–deuteron scattering process
Elastic scattering and breakup processes are constructed from two types of matrix elements containing the same initial state | φ with a deuteron and a free nucleon with the relative momentum q0
The final state φ | corresponds to a deuteron and a free nucleon with the relative momentum q0 and in the breakup, the final state φ0 | describes three free particles
Summary
A general operator form of the three-nucleon (3N ) scattering amplitude has potential applications in calculations that employ the so-called three dimensional (3D) formalism to calculate observables in the nucleon–deuteron scattering process. Solutions of the relevant equations using first order terms in the nucleon–nucleon transition operator were obtained in [1] and demonstrated that for certain kinematical configurations, the precision of the 3D calculations is better than the traditional partial wave approach This observation motivates the development of a full 3D calculation. The final state φ | corresponds to a deuteron and a free nucleon with the relative momentum q0 and in the breakup, the final state φ0 | describes three free particles For the latter, observables are constructed from. If each component of the momentum vectors is discretized over a grid of 32 points, the numerical representation of the scattering amplitude would require ≈ 1015 complex numbers. We briefly discuss the operator form of T | φ following the considerations in [2]
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