The Hyperspherical Harmonics Method: A Tool for Testing and Improving Nuclear Interaction Models

Abstract

The Hyperspherical Harmonics (HH) method is one of the most accurate techniques to solve the quantum mechanical problem for nuclear systems with A ≤ 4. In particular, by applying the Rayleigh-Ritz or Kohn variational principle, both bound and scattering states can be addressed, using either local or non-local interactions. Thanks to this versatility, the method can be used to test the two-and three-nucleon components of the nuclear interaction.In the present review we introduce the formalism of the HH method, both for bound and scattering states. In particular, we describe the implementation of the method to study the A = 3 and 4 scattering problem. Second, we present a selected choice of results of the last decade, most representative of the latest achievements. Finally, we conclude with a discussion of what we believe will be the most significant developments within the HH method for the next five-to-ten years.on Monte Carlo techniques, as the variational Monte Carlo (VMC) or the Green's function Monte Carlo (GFMC) methods (see Ref. [7], and references therein). There are then the methods linked to the shell model, as the no-core shell model (NCSM) or the realistic shell model (RSM), see Refs. [8] and [9,10], respectively. All these methods are quite powerful to study medium-mass nuclear bound states, but less accurate, apart from the GFMC and NCSM, for very light nuclei, as those with A = 3, 4. Furthermore, their extension to the scattering systems is not so trivial, and, in some cases, still not at reach.Restricting ourselves to the A = 3, 4 nuclear systems, both bound and scattering states, we have at hand very few accurate ab-initio methods, i.e. the Faddeev (Faddeev-Yakubovsky for A = 4) equations (FE) technique, solved in coordinate-or in momentum-space, the method based on the Alt-Grassberger-Sandhas (AGS) equations solved in momentum space, and the Hyperspherical Harmonics (HH) method presented here. We refer the reader to Refs. [11, 12] for the FE method in coordinate space, to Refs. [13,14] for the FE method in momentum space, to Refs. [15,16] for recent reviews on the AGS method. Clearly, each method has advantages and drawbacks. For instance, the FE method in momentum space can be applied to A = 3, 4 bound and scattering states in a wide energy range. However, the inclusion of the Coulomb interaction for charged particle scattering states is quite problematic. The FE method in coordinate space can handle the Coulomb interaction, but it has not yet been applied to scattering problems at very low-energy, and it has been applied only recently to study systems with larger A values [17]. It is though a method with in principle great possibilities of extension [12]. The AGS method, although working in momentum space, can handle the Coulomb interaction, and can be applied to a large variety of A = 3, 4 scattering states, in a wide energy range. However, the very low energy range, that of interest for nuclear astrophysics, i.e. below ∼ 100 keV, is still not accessible with the AGS method. The method has ...

Electric dipole moment of light nuclei in chiral effective field theory

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Implementation of Local Chiral Interactions in the Hyperspherical Harmonics Formalism

“…The hyperspherical harmonic method was firstly introduced in 1935 by Zernike and Brinkman [24], reintroduced later in the 60's by Delves [25], Simonov [26], Zickendraht [27], and Smith [28] and it is extensively applied nowadays to the study of fewbody systems. For recent reviews with applications to nuclear physics we refer the reader to the following references [29,30]. In this work, the hyperspherical harmonic functions are constructed to form irreducible representations of the SO(3) group of space rotations, the O(N) group of dynamical rotations in the space spanned by the N Jacobi vectors, and the S A permutation group of the A-particle system.…”

Comparative study of ${}^6$He $β$-decay based on different similarity-renormalization-group evolved chiral interactions

“…In this work, in an attempt to understand the origin of this discrepancy (possibly resulting in an unpremeditated effort to add to the existing confusion! ), we present a calculation of the GTME contributing to the β-decay of 6 He, within the hyperspherical-harmonics (HH) method developed by the Pisa group [6,7], and recently extended to deal with A = 6 nuclei [8]. The 6 He and 6 Li wave functions are obtained from a Hamiltonian including two-nucleon (2N ) interactions only.…”