A high-precision variational approach to three- and four-nucleon bound and zero-energy scattering states

Abstract

The hyperspherical harmonic (HH) method has been widely applied in recent times to the study of the bound states, using the Rayleigh-Ritz variational principle, and of low-energy scattering processes, using the Kohn variational principle, of A = 3 and 4 nuclear systems. When the wave function of the system is expanded over a sufficiently large set of HH basis functions, containing or not correlation factors, quite accurate results can be obtained for the observables of interest. In this paper, the main aspects of the method are discussed together with its application to the A = 3 and 4 nuclear bound and zero-energy scattering states. Results for a variety of nucleon-nucleon (NN) and three-nucleon (3N) local or non-local interactions are reported. In particular, NN and 3N interactions derived in the framework of the chiral effective field theory and NN potentials from which the high momentum components have been removed, as recently presented in the literature, are considered for the first time within the context of the HH method. The purpose of this paper is two-fold. First, to present a complete description of the HH method for bound and scattering states, including also detailed formulas for the computation of the matrix elements of the NN and 3N interactions. Second, to report accurate results for bound and zero-energy scattering states obtained with the most commonly used interaction models. These results can be useful for comparison with those obtained by other techniques and are a significant test for different future approaches to such problems. of the two-pion-exchange 3N interaction plus a phenomenological repulsive term. The strengths of the two contributions are adjusted to reproduce the triton binding energy and the nuclear matter equilibrium density, in conjunction with one of the Argonne NN potentials. In particular, the so-called Urbana IX 3N interaction [14] is often used with the Argonne AV18 NN interaction. A more recent and sophisticated 3N interaction model [15], still derived by the Urbana group, contains two-pion-exchange terms due to pion-nucleon scattering in S-and P -waves, three-pion-exchange terms due to ring diagrams with one ∆ resonance in the intermediate states, and again a phenomenological repulsive term. The model has five parameters which are fitted to the light nuclei mass spectrum. This more recent model has a rather complicated operatorial structure and is currently object of study.Another family of models for the 3N interaction is known as the Tucson-Melbourne [16] (TM) potential, which arises from an off-mass-shell model for the pionnucleon scattering based upon current algebra and a dispersion-theoretical axial vector amplitude dominated by the ∆ resonance. The model contains monopole form factors, whose cutoff is adjusted to reproduce the triton binding energy. More recently, the model has been revisited within a chiral symmetry approach [17], and it has been demonstrated that the contact term present in the TM model should be dropped. This new TM potential, known as TM ′ ...

Minimally nonlocal nucleon-nucleon potentials with chiral two-pion exchange includingΔresonances

“…It should be fairly straightforward to incorporate it in the few-nucleon calculations based on hyperspherical-harmonics expansion techniques favored by the Pisa group [66], or in the quantum Monte Carlo ones preferred by the ANL/ASU/JLab/LANL collaboration [18]. The Fortran computer program generating the potential will be made available upon request.…”

Tuning the 3Nforce from 3Nscattering data

“…The three-body problem was solved with the Hyperspherical Harmonics method, reviewed in Ref. [6]. The p − d scattering wave function with relative orbital angular momentum L , total spin S and total angular momentum J, is written as the sum of an internal and an asymptotic part…”

Comparative study of three-nucleon force models in nuclear matter

“…Within the BHF approach the defect function should be calculated selfconsistently with the G matrices (17) and the single-particle potentials (19). Thus the average effective two-body force (21) should be calculated self-consistently and added to the bare NN force at each iterative step of the calculations. To simplify the numerical calculations and following [50,51], in the present work we use central correlation functions g(i,j ) independent on spin and isospin.…”