Antisymmetrisation in a Jacobi coordinate based no-core shell model approach

Abstract

The theory of the strong interaction, Quantum Chromodynamics (QCD), provides the generally accepted description of strongly interacting processes. Due to the confinement of the fundamental degrees of freedom of QCD, the quarks and gluons, the objects that are observable in experiments are their bound states, the hadrons. Since QCD cannot be analysed within perturbation theory in the low-energy domain relevant here, different approaches are necessary. One of these, an effective field theory (EFT) approach, has been very successful for the description of low-energy reactions. Especially, systems of few nucleons, which are the subject of this work, require a systematic scheme for the formulation of the interactions of the nucleons involved to be able to predict their low-energy observables with controlled accuracy. Here, a brief summary of the content of this thesis is given:<br /> In Chapter 1, we constitute the problem that we want to advance, namely the calculation of bound and excited states of few-nucleon systems by solving the non-relativisic Schrödinger equation. We motivate, why a systematic derivation of the nuclear interactions is needed, and review the theoretical framework of chiral EFT. The expansion of the chiral nuclear potential and the appearance of few-nucleon forces up to fourth order is discussed. Additionally, a decoupling of high- and low- momenta contributions is desirable for the potential, realised by renormalisation group methods that are introduced briefly. At the end of this chapter, we give an overview of existing techniques for the numerical calculation of light nuclei, their fields of applicability and strengths and failures. Special emphasis lies on the no-core shell model (NCSM) approach since it is the aim of this thesis to develop a Jacobi coordinate based formalism within this method. Jacobi or relative coordinates are more advisable for calculations that explicitly take higher order contributions to the nuclear potential into account. <br /> In the NCSM, the few-nucleon basis states are given in a harmonic oscillator basis. Due to the Pauli principle, the basis states of bound few-nucleon systems have to be antisymmetric. An algorithm for the antisymmetrisation of the Jacobi coordinate based states is derived in Chapter 2 which forms a matrix eigenvalue problem. We describe this formalism in detail for three-nucleon states and generalise the explicit evaluation of the required matrix elements to systems with <i>A </i>nucleons. To solve the Schrödinger equation, new basis states are derived which make it possible to consider three- and few-nucleon contributions explicitly. With this, the solution of the Schrödinger equation, also a matrix eigenvalue equation, takes a similar form as the antisymmetrisation of basis states. <br /> In Chapter 3, numerical aspects of the calculations are discussed including tests of the parallelisation. The parallelisation is strongly required due to the large dimensions of the matrices for the antisymmetrisation and the Schrödinger equation likewise. Additionally, we specify the evaluation of the matrix elements of the two-nucleon contribution to the nuclear potential and the method with which we analyse the results. <br /> The results for the binding energies of ³H and ⁴He as well as for the ground state, the lowest excited state and of the excitation energy of ⁶Li considering NN forces are presented in Chapter 4. We also discuss first results of the binding energy of ⁷Li and our expectations for larger model spaces that are in progress. Furthermore, we analyse these results with regard to their convergence and their cutoff dependence.<br /> Finally, a summary and an outlook on future work is given in Chapter 5.<br /> The Appendices contain technical details and explicit calculations.