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Register a new userProgram to calculate coefficients of transformations between three-particle hyperspherical harmonics
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Authors
Victor D. Efros
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A program to calculate the three-particle hyperspherical brackets is presented. Test results are listed and it is seen that the program is well applicable up to very high values of the hypermomentum and orbital momenta. The listed runs show that it is also very fast. Applications of the brackets to calculating interaction matrix elements and constructing hyperspherical bases for identical particles are described. Comparisons are done with the programs published previously.
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A program is created to compute recursively the Moshinsky brackets. It is very fast and provides highly accurate results. In the case of the double-precision computations with a single-processor consumer notebook, the computing time per bracket at any not small oscillator excitations is on the scale of 10^{-8} s and the accuracy is very good for the total number of quanta up to 80. The program is easy to handle.
The hyperspherical harmonic basis is used to describe bound states in an $A$--body system. The approach presented here is based on the representation of the potential energy in terms of hyperspherical harmonic functions. Using this representation, the matrix elements between the basis elements are simple, and the potential energy is presented in a compact form, well suited for numerical implementation. The basis is neither symmetrized nor antisymmetrized, as required in the case of identical particles; however, after the diagonalization of the Hamiltonian matrix, the eigenvectors reflect the symmetries present in it, and the identification of the physical states is possible, as it will be shown in specific cases. We have in mind applications to atomic, molecular, and nuclear few-body systems in which symmetry breaking terms are present in the Hamiltonian; their inclusion is straightforward in the present method. As an example we solve the case of three and four particles interacting through a short-range central interaction and Coulomb potential.
We compute two-particle production in p+A collisions and extract azimuthal harmonics, using the dilute-dense formalism in the Color Glass Condensate framework. The multiple scatterings of the partons inside the projectile proton on the dense gluons inside the target nucleus are expressed in terms of Wilson lines. They generate interesting correlations, which can be partly responsible for the signals of collectivity measured at RHIC and at the LHC. Most notably, while gluon Wilson loops yield vanishing odd harmonics, quark Wilson loops can generate sizable odd harmonics for two particle correlations. By taking both quark and gluon channels into account, we find that the overall second and third harmonics lie rather close to the recent PHENIX data at RHIC.
The Hyperspherical Harmonics basis, without a previous symmetrization step, is used to calculate binding energies of the nuclear A=6 systems using a version of the Volkov potential acting only on s-wave. The aim of this work is to illustrate the use of the nonsymmetrized basis to deal with permutational-symmetry-breaking term in the Hamiltonian, in the present case the Coulomb interaction.
The non-symmetrized hyperspherical harmonics method for a three-body system, composed by two particles having equal masses, but different from the mass of the third particle, is reviewed and applied to the $^3$H, $^3$He nuclei and $^3_{Lambda}$H hyper-nucleus, seen respectively as $nnp$, $ppn$ and $NNLambda$ three-body systems. The convergence of the method is first tested in order to estimate its accuracy. Then, the difference of binding energy between $^3$H and $^3$He due to the difference of the proton and the neutron masses is studied using several central spin-independent and spin-dependent potentials. Finally, the $^3_{Lambda}$H hypernucleus binding energy is calculated using different $NN$ and $Lambda N$ potential models. The results have been compared with those present in the literature, finding a very nice agreement.
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