The Two-Nucleon and Three-Nucleon System in Three Dimensions

Abstract

We present a brief overview of the three-dimensional formalism that is under development in our group. Using the 3D momentum eigenstates of the nucleon directly, instead of relying on the partial wave decomposition of operators involved in the calculations, allows us to use a very direct approach. This in turn enabled us to successfully tackle a large variety of few-body problems. Our calculation of the two nucleon transition operator and bound state can incorporate a very general form of the two-nucleon potential. Calculations of the three-nucleon bound state can include in addition to the two-nucleon potential also a very general operator form of the threenucleon force. Recently the 3D formalism is also applied to processes that involve electro-weak probes. Carrying out these calculations for a wide spectrum of two-nucleon and three-nucleon potentials using the classical partial wave approach is unpractical due to the complicated spin structure of the operators. Using the 3D formalism, the calculations can be quickly adapted to test new models. The starting point for our calculations are operator forms of two-nucleon (2N) and three-nucleon (3N) operators and states, developed to take into account symmetry considerations. In these forms the operators and states are typically written as linear combinations of scalar functions and spin, isospin operators. Our rst goal is to rewrite the fundamental equations that govern a particular few-body problem into a linear problem acting in a space spanned by the scalar functions.The 2N potential has the general form [1]:Here p |V tmt |p is the matrix element of the 2N force between 2N relative momentum eigenstates |p , |p and states |tm t where the isospins of the two nucleons are coupled to t with projection m t . Finallyw i (p , p) are a spin operator and the potential can be reconstructed from a set of scalar functions v tmt i (|p |, |p|, p · p). The form (1) takes into account the partity and time reversal symmetries and assumes no isospin mixing. The same symmetry considerations that led to (1) can be applied to the transition operator satisfying the LippmannSchwinger equatioň * corresponding author; e-mail: kacper.topolnicki@uj.edu.pl where |p is the 3D eigenstate of the relative 2N momentum,σ(1) (σ (2)) is a vector of spin operators acting in the space of particle 1 (2), |1m d is a spin state with the spins of the particles coupled to the total spin 1 with the projection m d . Note that the isospin of the system is 0. The main challenge is to calculate the scalar functions {φ 1 , φ 2 } that can be used to reconstruct the 2N wave function. The general form of the 3N bound state is more complicated [3]. The wave function projected onto a 3N isospin state (t 1 2 )T | (where the isospins of two particles are coupled to t and then coupled with the isospin of the third particle to a total isospin T ) has the form(4) and a more detailed description of operators involved in (4) can be found in [3]. Also in this case the 3N wave function can be reconstructed from the set of ...