2N and 3N Systems in a Three Dimensional Formalism

K Topolnicki ,A Nogga ,H Witała ,R Skibiński ,J Golak ,A E Elmeshneb ,H Kamada

doi:10.1007/s00601-013-0793-z

Abstract

We present an overview of our framework used to treat two- and three-nucleon (2N, 3N) systems employing three dimensional momentum eigenstates. Using a three dimensional formalism instead of the classical partial wave approach is an attractive alternative for a number of reasons, the most prominent being the very direct way of performing calculations. With the use of our tools it is possible to produce a working numerical realization of calculations in only a couple of steps from the fundamental (Schrodinger or Lippmann–Schwinger) equations. The FORTRAN implementations of the most complicated parts of the calculations are generated automatically by $${Mathematica^{\circledR}}$$ software that was written in our group. Additionally, at higher energies, three dimensional calculations avoid problems arising from slow convergence of partial wave decomposition based techniques. Our approach utilizes a very general form of the 2N and 3N forces and has been successfully used to obtain results for the 2N transition operator as well as for the 2N and 3N bound states (Golak et al. in Phys Rev C 81:034006, 2010; Few-Body Syst 53:237, 2012a; Few-Body Syst, 2012b).

Highlights

A numerical realization of calculations involving the 2N and 3N bound state and the transition operator is classically achieved by using partial wave projected operators
We explore a different approach that involves three dimensional momentum vectors
The possibility of encapsulating the most complicated parts of the calculation in an automatically generated code allows us to create a numerical realization that is very directly related to the fundamental equations

Summary

Introduction

A numerical realization of calculations involving the 2N and 3N bound state and the transition operator is classically achieved by using partial wave projected operators. The possibility of encapsulating the most complicated parts of the calculation in an automatically generated code allows us to create a numerical realization that is very directly related to the fundamental equations. Permutations within the spin (isospin) space are explicitly represented as 4 × 4 (for 2N systems) or 8 × 8 (for 3N systems) dimensional matrices. These properties, together with momentum space permutations Additional optimization is performed afterwards by the FORTRAN compiler Using these tools it is convenient to perform the most complex parts of the calculation

Formal Ingredients

Numerical Treatment of Large Linear Operators

Results and Summary