Three-nucleon bound state calculations using the three dimensional formalism

Kacper Topolnicki,Alessandro Grassi,Roman Skibiński,Jacek Golak,Volodymyr Soloviov,Yuriy Volkotrub,Henryk Witała,A Wrońska,W Przygoda,A Magiera

doi:10.1051/epjconf/201919905021

Kacper Topolnicki, Alessandro Grassi + Show 8 more

Open Access

https://doi.org/10.1051/epjconf/201919905021

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Abstract

The traditional method of carrying out few-nucleon calculations is based on the angular momentum decomposition of operators relevant to the calculation. Expressing operators using a finite-sized partial wave basis enables the calculations to be carried out using a small amount of numerical work. Unfortunately, certain calculations that involve higher energies or long range potentials, require including a large number of partial waves in order to get converged results. This is problematic because such an approach requires a numerical implementation of heavily oscillating functions. Modern computers made it possible to carry out few-nucleon calculations without using angular momentum decomposition and instead to work directly with the three dimensional degrees of freedom of the nucleons. In this paper we briefly describe the, so called 3D approach and present preliminary results related to the 3He bound state obtained within this formalism.

Highlights

The starting point of our bound state calculations is the Faddeev equation, for the Faddeev component of the three-nucleon (3N) wave function | ψ :| ψ = G0(E) V 12N + V 13N 1 + P | ψ (1)where E is the bound state energy, G0(E) is the free propagator, Vi2N is the two nucleon (2N) potential acting between particles j and k (i j k i) and Vi3N is a part of the 3N potential that is symmetric with respect to the exchange of particles j and k (i j k i)
| ψ = G0(E) V 12N + V 13N 1 + P | ψ where E is the bound state energy, G0(E) is the free propagator, Vi2N is the two nucleon (2N) potential acting between particles j and k (i j k i) and Vi3N is a part of the 3N potential that is symmetric with respect to the exchange of particles j and k (i j k i)
In practical numerical calculations these vectors are represented as multidimensional arrays and the implementation of the Arnoldi algorithm contains parallelized subroutines that add vectors, apply linear operators and compute scalar products

Summary

Introduction

The starting point of our bound state calculations is the Faddeev equation, for the Faddeev component of the three-nucleon (3N) wave function | ψ : The full 3N bound state | Ψ can be reconstructed from the Faddeev component using:

Results

Conclusion