On the Method of N-Body Hyperspherical Basis Symmetrization

Abstract

Parentage Scheme of Summarization (PSS) to the N-body symmetrized basis construction necessary for the description of the structural characteristics and decay reactions of the hypernuclear and nuclear systems with arbitrary amount of particles is introduced. Proposed method allows to construct N-particle symmetrized hyperspherical functions on the bases of Nparticle hyperspherical functions symmetrized with respect to N−1-particles by the use of the Kinematic Rotation Coefficients (KRC) related with the (N-1)-th and N-th particle permutations. The main problem that arises when investigating dynamics of few-body systems in physics is the problem of kinematic rotations under particle permutations. When number of particles increases, kinematic rotations include not only particle permutation but also transitions between different possible configurations, and mathematical calculations using complex general formula become impossible. Moreover, no general formula exists for systems with more than four particles. In order to solve kinematic rotation problem for N-particle systems, the recurrence method of determination of the KRC is applied. According to this method, the initial coefficients with the lowest quantum numbers are calculated by solving the overlap integral analytically, wave functions with arbitrary quantum numbers are expanded in terms of the basic hyperspherical functions, and the kinematic rotations of the obtained expansion are performed with the use of already known coefficients with the lowest quantum numbers. Significant advantage of the recurrence method is that no principal difficulties arise when increasing number of particles. Furthermore, recurrence relations contain numerical coefficients that are easy to evaluate by substituting appropriate quantum numbers. The problem of symmetrization of N-Body hyperspherical functions is solved by the use of the PSS. A construction scheme is given for the symmetrized hyperspherical basis in (4+1) and (5+1) configurations. The relations between parentage coefficients and the KRC of the basic hyperspherical functions are obtained. Parentage coefficients for N-Body systems are introduced. It is demonstrated that no principal difficulties arise when increasing number of particles.

Six-body calculations for hyperhydrogenΛ6H

“…6 Λ H hypernucleus consist of five identical particles and  hyperon, the symmetrization with respect to five particles may be done easily with the use of five body symmetrization coefficients [9], because coordinates from x, y, z, w, u (1) coincide with five particle Jacobi coordinates. For sixbody a construction scheme of the symmetrized HH basis for (5+1) configuration is given [8]: …”

Microscopic investigation of the structure characteristics and wave functions of the five-body hypernucleusHeλ5

“…Five-body hyperspherical functions Ψ ఱ ర ሾ ̅ ሿ ഥ ఔ ഥ భమయ ర ሺ߱ሻ, symmetrized with respect to four identical particles, are constructed in (4+1) configuration by the use of the Parentage Scheme of Symmetrization [1]. In formula (7) ൣ݂ ̅ ൧ represents a Yong scheme of the irreducible representation of the group ܵ ସ and is obtained from the Young scheme corresponding to the five-body function Ψ ఱ ሾሿሺሻఔ by removing the square corresponding to the fifth particle, ݉ ഥ denotes the rows of the ൣ݂ ̅ ൧…”

“…representation, ν is the number of ൣ݂ ̅ ൧ with given K and L, ‫ܥ‬ ర భమయ ሾ̅ ሿ ഥ ఔ ഥ are four particle symmetrization coefficients obtained in [1].…”

“…Generalized mathematical formalism to the construction of N-particle fully symmetrized hyperspherical functions on the basis of the N-particle hyperspherical functions symmetrized with respect to N−1-particles was introduced in [1]. The microscopic approach to the study of the structure and mesonic decays of four-body hypernucleus ‫ܪ‬ ఒ ସ by the use of the HFM in momentum space was developed and structure characteristics and decay rates were obtained [2][3][4][5].…”

“…The Parentage Scheme of Summarization to the N-body symmetrized basis construction [1], necessary for the description of the structural characteristics and decay reactions of the hypernuclear and nuclear systems with arbitrary amount of particles, is applied to the solution of five-body problem in hypernuclear physics. Hypernucleus ‫݁ܪ‬ ఒ ହ as a system of four nucleons and one hyperon is investigated by the use of the Hyperspherical Function Method in momentum space.…”