Four-Body Calculation of Proton-He3Scattering

Abstract

The four-body equations of Alt, Grassberger and Sandhas are solved, for the first time, for proton-3 He scattering including the Coulomb interaction between the three protons using the method of screening and renormalization as it was done recently for proton-deuteron scattering. Various realistic two-nucleon potentials are used. Large Coulomb effects are seen on all observables. Comparison with data at different energies shows large deviations in the proton analyzing power but quite reasonable agreement in other observables. The effect of nucleon-nucleon magnetic moment interaction and correlations between p-d and p-3 He analyzing powers are studied.PACS numbers: 21.30. -x, 21.45.+v, 24.70.+s, 25.10.+s Modern calculations of light nuclear systems A ≤ 12 are essential to our understanding of the force models that have been developed to describe how nucleons interact at low energies [1,2]. Of these nuclear systems, the four-nucleon (4N ) system is particularly important because it gives rise, experimentally, to the simplest set of nuclear reactions that shows the complexity of heavier systems and the Coulomb interaction manifest itself in new ways relative to what is observed in the threenucleon (3N ) system. Theoretically it is also important because with powerful numerical techniques and fast computers one can calculate not only bound state properties [3] but also scattering observables [4,5,6,7,8,9, 10] for a number of elastic, transfer and breakup reactions that place new challenges to our understanding of the underlying force models. The importance of scattering calculations also has to do with the possibility to probe states in the continuum associated with specific resonances, states of higher angular momentum than corresponding bound states, effects that depend on the spin orientation of the projectile and/or target, and threshold effects on the observables, among others.While the three-nucleon system has been extensively studied [11,12] through neutron-deuteron (nd) and proton-deuteron (pd) elastic scattering and breakup experiments, exact calculations using realistic force models as well as interactions derived from Effective Field Theory were restricted, for a long time, to the nd system due to limitations in including the Coulomb force in the description of pd scattering beyond low energy pd → pd and pd ↔ γ 3 He calculations [13,14] in the framework of the variational hyperspherical approach. The situation has now changed due to the work of Refs. [15,16] where calculations of pd → pd, pd → ppn, pd ↔ γ 3 He, γ 3 He → ppn, e 3 He → e ′ pd, and e 3 He → e ′ ppn were performed at energies ranging from 1 MeV in the center of mass (c.m.) system to the pion production threshold. The work is based on the solution of the momentum-space Alt, Grassberger and Sandhas (AGS) equations [17] together with the screening and renormalization approach [18,19,20] for the Coulomb interaction leading to the results of observables that are independent of the screening radius, provided it is sufficiently large.In the prese...

Benchmark calculation ofp−H3andn−He3

“…Very accurate solutions of the 4N scattering problem using the AGS technique [14][15][16] were obtained already a few years ago. The long-range Coulomb interaction in this approach is taken into account using the screening and renormalization method [17,18].…”

“…The AGS equations [13] for the four-body transition operators were derived assuming short-range interactions, but together with the screening and renormalization method [15,17,18], they can be applied also to systems with repulsive Coulomb force. The isospin formalism enables the symmetrization of the AGS equations [14] in the 4N system, where there are only two distinct four-particle partitions, one of the 3 + 1 type and one of the 2 + 2 type, denoted by α = 1 and 2, respectively.…”

Integral equations for the four-body problem

“…As a consequence, the tetramer states predicted in [29] that have (negative) energies larger than the trimer ground state energy are actually resonances rather than genuine stationary states. In the nuclear physics context, however, exact numerical solutions of the four-body problem with real space interaction potentials V (r ij ) can be obtained thanks to the reformulation in terms of momentum-space Faddeev-Yakubovsky integral equations [30,31]. A numerical solution of a low-energy effective theory based on this integral formulation was used in Ref.…”

Reactions in the four-nucleon system above breakup threshold

“…However, their numerical solution using realistic force models only became possible many years later. In the last decade accurate numerical calculations for low-energy nucleon-trinucleon elastic scattering have been performed using both coordinate-space and momentum-space rigorous approaches, namely, the hyperspherical harmonics (HH) expansion method [4,5], the Faddeev-Yakubovsky (FY) equations [1] for the wave function components [6], and the Alt, Grassberger and Sandhas (AGS) equations [3] for transition operators [7,8]. The reliability of all these methods was confirmed in a benchmark calculation [9] for neutron-3 H (n-3 H) and proton-3 He (p-3 He) elastic scattering observables.…”