The µ − + 2 H → ν µ +n+n, µ − + 3 He → ν µ + 3 H, µ − + 3 He → ν µ +n+d and µ − + 3 He → ν µ +n+n+p capture reactions are studied with various realistic potentials under full inclusion of final state interactions. Our results for the two-and three-body break-up of 3 He are calculated with a variety of nucleon-nucleon potentials, among which is the AV18 potential, augmented by the Urbana IX three-nucleon potential. Most of our results are based on the single nucleon weak current operator. As a first step, we have tested our calculation in the case of the µ − + 2 H → ν µ + n + n and µ − + 3 He → ν µ + 3 H reactions, for which theoretical predictions obtained in a comparable framework are available. Additionally, we have been able to obtain for the first time a realistic estimate for the total rates of the muon capture reactions on 3 He in the break-up channels: 544 s −1 and 154 s −1 for the n + d and n + n + p channels, respectively. Our results have also been compared with the most recent experimental data, finding a rough agreement for the total capture rates, but failing to reproduce the differential capture rates.
We compare results from traditional partial wave treatment of deuteron electro-disintegration with a new approach that uses three-dimensional formalism. The new framework for the two-nucleon (2N) system using a complete set of isospin-spin states made it possible to construct simple implementations that employ a very general operator form of the current operator and 2N states.
Abstract. The μ − + 3 He → ν μ + n + d and μ − + 3 He → ν μ + n + n + p capture reactions are studied under full inclusion of final state interactions with the AV18 nucleon-nucleon potential, augmented by the Urbana IX three-nucleon force, and employing the single nucleon weak current operator. We give first realistic estimates of the total capture rates: 544 s −1 and 154 s −1 for the n + d and n + n + p channels, respectively. Our results are compared with the most recent experimental data, finding a rough agreement for the total capture rates, but failing to reproduce the differential capture rates.
Abstract. We present a three-dimensional (3D) description of muon induced deuteron disintegration. This reaction is treated as the decay of the muonic atom with the muon initially on the lowest K shell. Our aim is to calculate the total and differential decay rates. We work in momentum space and use 3D momentum eigenstates directly. This approach allowed us to calculate the appropriate nuclear matrix elements, necessary building blocks for the differential decay rate, in a single step. For contrast -in classical calculations many partial-waves have to be taken into account. We achieved a very good agreement between the 3D and partial-wave methods for calculations that involve single-nucleon currents. Our result for the total decay rate is also in agreement with experimental values, though these are not very precise. This success motivates us to also include two-nucleon current contributions that include the meson exchange currents. Additionally, our formalism can also be applied to other, so far poorly described, processes like: μ + 3 He → ν + n + d or μ + 3 He → ν + n + n + p.
We present an overview of our framework used to treat two-and three-nucleon (2N, 3N) systems employing three dimensional momentum eigenstates. Using a three dimensional formalism instead of the classical partial wave approach is an attractive alternative for a number of reasons, the most prominent being the very direct way of performing calculations. With the use of our tools it is possible to produce a working numerical realization of calculations in only a couple of steps from the fundamental (Schrödinger or Lippmann-Schwinger) equations. The FORTRAN implementations of the most complicated parts of the calculations are generated automatically by Mathematica ® software that was written in our group. Additionally, at higher energies, three dimensional calculations avoid problems arising from slow convergence of partial wave decomposition based techniques. Our approach utilizes a very general form of the 2N and 3N forces and has been successfully used to obtain results for the 2N transition operator as well as for the 2N and 3N bound states (Golak et al. in Phys Rev C 81:034006, 2010; Few-Body Syst 53:237, 2012a; Few-Body Syst, 2012b).
Abstract.We plan to investigate the role of meson exchange currents in the description of the μ − + d → ν μ + n + n and μ − + 3 He → ν μ + 3 H reactions. They both are treated as the decay of the corresponding muonic atoms, with the muon initially on the lowest K shell. The muon binding energy in these atoms can be safely neglected and in the initial state we deal essentially with the deuteron (or 3 He) and muon at rest. These two reactions are interesting for several reasons. First of all, they offer a testing ground for the nuclear wave functions, which for any nucleon-nucleon (NN) and three-nucleon (3N) forces can be constructed for such light systems with great accuracy. In these reactions few-nucleon weak current operators are an important dynamical ingredient. In the current operators apart from the relatively well known single nucleon contributions, two-nucleon parts (generated by various meson exchanges) play an important role. Their details are not well known and several models should be considered. We present our formalism for dealing with these reactions and a simple method for partial wave decomposition of the two-nucleon operators. The crucial nuclear matrix elements of the corresponding weak current operators will be calculated in the momentum space and using partial wave decomposition. The effect of meson exchanges will be investigated in the energy spectrum of the emitted neutrinos (in the deuteron case) and in the total decay rates for the two reactions. We will employ various models of NN and 3N forces, such as the Bonn B or chiral NNLO potentials. Our results with the single nucleon currents look already very promising and we hope for the improvement in the description of the experimental data, when dominant two-nucleon current operators are included in our framework.
We present a brief overview of the three-dimensional formalism that is under development in our group. Using the 3D momentum eigenstates of the nucleon directly, instead of relying on the partial wave decomposition of operators involved in the calculations, allows us to use a very direct approach. This in turn enabled us to successfully tackle a large variety of few-body problems. Our calculation of the two nucleon transition operator and bound state can incorporate a very general form of the two-nucleon potential. Calculations of the three-nucleon bound state can include in addition to the two-nucleon potential also a very general operator form of the threenucleon force. Recently the 3D formalism is also applied to processes that involve electro-weak probes. Carrying out these calculations for a wide spectrum of two-nucleon and three-nucleon potentials using the classical partial wave approach is unpractical due to the complicated spin structure of the operators. Using the 3D formalism, the calculations can be quickly adapted to test new models. The starting point for our calculations are operator forms of two-nucleon (2N) and three-nucleon (3N) operators and states, developed to take into account symmetry considerations. In these forms the operators and states are typically written as linear combinations of scalar functions and spin, isospin operators. Our rst goal is to rewrite the fundamental equations that govern a particular few-body problem into a linear problem acting in a space spanned by the scalar functions.The 2N potential has the general form [1]:Here p |V tmt |p is the matrix element of the 2N force between 2N relative momentum eigenstates |p , |p and states |tm t where the isospins of the two nucleons are coupled to t with projection m t . Finallyw i (p , p) are a spin operator and the potential can be reconstructed from a set of scalar functions v tmt i (|p |, |p|, p · p). The form (1) takes into account the partity and time reversal symmetries and assumes no isospin mixing. The same symmetry considerations that led to (1) can be applied to the transition operator satisfying the LippmannSchwinger equatioň * corresponding author; e-mail: kacper.topolnicki@uj.edu.pl where |p is the 3D eigenstate of the relative 2N momentum,σ(1) (σ (2)) is a vector of spin operators acting in the space of particle 1 (2), |1m d is a spin state with the spins of the particles coupled to the total spin 1 with the projection m d . Note that the isospin of the system is 0. The main challenge is to calculate the scalar functions {φ 1 , φ 2 } that can be used to reconstruct the 2N wave function. The general form of the 3N bound state is more complicated [3]. The wave function projected onto a 3N isospin state (t 1 2 )T | (where the isospins of two particles are coupled to t and then coupled with the isospin of the third particle to a total isospin T ) has the form(4) and a more detailed description of operators involved in (4) can be found in [3]. Also in this case the 3N wave function can be reconstructed from the set of ...
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