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  is often used with the Argonne AV18 NN interaction. A more recent and sophisticated 3N interaction model , 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  (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 , and it has been demonstrated that the contact term present in the TM model should be dropped. This new TM potential, known as TM ′ ...
Spurred by the recent complete determination of the weak currents in two-nucleon systems up to O(Q 3 ) in heavy-baryon chiral perturbation theory, we carry out a parameter-free calculation of the threshold S-factors for the solar pp (proton-fusion) and hep processes in an effective field theory that combines the merits of the standard nuclear physics method and systematic chiral expansion. The power of the EFT adopted here is that one can correlate in a unified formalism the weak-current matrix elements of two-, three-and four-nucleon systems. Using the tritium β-decay rate as an input to fix the only unknown parameter in the theory, we can evaluate the threshold S factors with drastically improved precision; the results are Spp(0) = 3.94×(1 ± 0.004)×10 −25 MeV-b and S hep (0) = (8.6 ± 1.3)×10 −20 keV-b. The dependence of the calculated S-factors on the momentum cutoff parameter Λ has been examined for a physically reasonable range of Λ. This dependence is found to be extremely small for the pp process, and to be within acceptable levels for the hep process, substantiating the consistency of our calculational scheme.
Different models for conserved two-and three-body electromagnetic currents are constructed from two-and three-nucleon interactions, using either meson-exchange mechanisms or minimal substitution in the momentum dependence of these interactions. The connection between these two different schemes is elucidated. A number of low-energy electronuclear observables, including (i) np radiative capture at thermal neutron energies and deuteron photodisintegration at low energies, (ii) nd and pd radiative capture reactions, and (iii) isoscalar and isovector magnetic form factors of 3 H and 3 He, are calculated in order to make a comparative study of these models for the current operator. The realistic Argonne v18 two-nucleon and Urbana IX or Tucson-Melbourne three-nucleon interactions are taken as a case study. For A=3 processes, the bound and continuum wave functions, both below and above deuteron breakup threshold, are obtained with the correlated hyperspherical-harmonics method. Three-body currents give small but significant contributions to some of the polarization observables in the 2 H(p, γ) 3 He process and the 2 H(n, γ) 3 H cross section at thermal neutron energies. It is shown that the use of a current which did not exactly satisfy current conservation with the two-and three-nucleon interactions in the Hamiltonian was responsible for some of the discrepancies reported in previous studies between the experimental and theoretical polarization observables in pd radiative capture.
PUBLISHED VERSIONSchiavilla, R.; Stocks, V. G.; Glockle, W.; Kamada, H.; Nogga, A.; Carlson, J.; Machleidt, R.; Pandharipande, V. R.; Wiringa, R. B.; Kievsky, A.; Rosati, S.; Viviani, M. Weak capture of protons by protons Physical Review C, 1998; 58(2) The cross section for the proton weak capture reaction 1 H(p,e ϩ e ) 2 H is calculated with wave functions obtained from a number of modern, realistic high-precision interactions. To minimize the uncertainty in the axial two-body current operator, its matrix element has been adjusted to reproduce the measured Gamow-Teller matrix element of tritium ␤ decay in model calculations using trinucleon wave functions from these interactions. A thorough analysis of the ambiguities that this procedure introduces in evaluating the two-body current contribution to the pp capture is given. Its inherent model dependence is in fact found to be very weak. The overlap integral ⌳ 2 (Eϭ0) for the pp capture is predicted to be in the range 7.05-7.06, including the axial two-body current contribution, for all interactions considered. ͓S0556-2813͑98͒06908-8͔ PACS number͑s͒: 21.30. Ϫx, 21.45.ϩv, 25.10.ϩs, 95.30.Cq
In the past, several efficient methods have been developed to solve the Schrödinger equation for four-nucleon bound states accurately. These are the Faddeev-Yakubovsky, the coupled-rearrangement-channel Gaussian-basis variational, the stochastic variational, the hyperspherical variational, the Green's function Monte Carlo, the no-core shell model and the effective interaction hyperspherical harmonic methods. In this article we compare the energy eigenvalue results and some wave function properties using the realistic AV8 NN interaction. The results of all schemes agree very well showing the high accuracy of our present ability to calculate the four-nucleon bound state. 21.45.+v, 24
Abstract.A variational technique to describe the ground and scattering states below the break-up threshold for a three-nucleon system is developed. The method consists in expanding the wave function in terms of correlated Harmonic Hyperspherical functions suitable to handle the large repulsion contained in the nuclear potential at short distances; three body forces have also been considered. The inclusion of the pp Coulomb repulsion in the p-d processes does not cause any particular problem, since no partial wave decomposition of the interaction is performed. Accurate numerical results are given for ground state properties and scattering lenghts, phase shifts and mixing parameters at three different energies of the incident nucleon. The agreement with other available results and with experimental analyses is higlly satisfactory.
We present fully local versions of the minimally non-local nucleon-nucleon potentials constructed in a previous paper [M. Piarulli et al., Phys. Rev. C 91, 024003 (2015)], and use them in hypersperical-harmonics and quantum Monte Carlo calculations of ground and excited states of 3 H, 3 He, 4 He, 6 He, and 6 Li nuclei. The long-range part of these local potentials includes oneand two-pion exchange contributions without and with ∆-isobars in the intermediate states up to order Q 3 (Q denotes generically the low momentum scale) in the chiral expansion, while the short-range part consists of contact interactions up to order Q 4 . The low-energy constants multiplying these contact interactions are fitted to the 2013 Granada database in two different ranges of laboratory energies, either 0-125 MeV or 0-200 MeV, and to the deuteron binding energy and nn singlet scattering length. Fits to these data are performed for three models characterized by long-and short-range cutoffs, R L and R S respectively, ranging from (R L , R S ) = (1.2, 0.8) fm down to (0.8, 0.6) fm. The long-range (short-range) cutoff regularizes the one-and two-pion exchange (contact) part of the potential.
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