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 ′ ...
A two-nucleon potential and consistent electromagnetic currents are derived in chiral effective field theory (χ EFT) at, respectively, Q 2 (or N 2 LO) and eQ (or N 3 LO), where Q generically denotes the low-momentum scale and e is the electric charge. Dimensional regularization is used to renormalize the pion-loop corrections. A simple expression is derived for the magnetic dipole (M1) operator associated with pion loops, consisting of two terms, one of which is determined, uniquely, by the isospin-dependent part of the two-pion-exchange potential. This decomposition is also carried out for the M1 operator arising from contact currents, in which the unique term is determined by the contact potential. Finally, the low-energy constants entering the N 2 LO potential are fixed by fits to the np Sand P -wave phase shifts up to 100 MeV laboratory energies.
The objectives of the present work are twofold. The first is to address and resolve some of the differences present in independent, chiral-effective-field-theory (χEFT) derivations up to one loop, recently appeared in the literature, of the nuclear charge and current operators. The second objective is to provide a complete set of χEFT predictions for the structure functions and tensor polarization of the deuteron, for the charge and magnetic form factors of 3 He and 3 H, and for the charge and magnetic radii of these few-nucleon systems. The calculations use wave functions derived from high-order chiral two-and three-nucleon potentials and Monte Carlo methods to evaluate the relevant matrix elements. Predictions based on conventional potentials in combination with χEFT charge and current operators are also presented. There is excellent agreement between theory and experiment for all these observables for momentum transfers up to q 2.0-2.5 fm −1 ; for a subset of them, this agreement extends to momentum transfers as high as q ≃ 5-6 fm −1 . A complete analysis of the results is provided.
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.
The electromagnetic charge operator in a two-nucleon system is derived in chiral effective field theory (χEFT) up to order e Q (or N4LO), where Q denotes the low-momentum scale and e is the electric charge. The specific form of the N3LO and N4LO corrections from, respectively, onepion-exchange and two-pion-exchange depends on the off-the-energy-shell prescriptions adopted for the non-static terms in the corresponding potentials. We show that different prescriptions lead to unitarily equivalent potentials and accompanying charge operators. Thus, provided a consistent set is adopted, predictions for physical observables will remain unaffected by the non-uniqueness associated with these off-the-energy-shell effects.
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
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