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Australian Journal of Physics

Renormalisation of g-boson Effects in an s-d-g Boson Model

KA Amos, X.J. Chen, A.E.L. Dieperink, Iain Morrison

Published Oct 1, 1988• 4 citations
Publisher

Abstract

A renormalisation method is developed in which g-boson coupling effects in a transformed Hamiltonian are minimised. The unitary transformation upon which this is based is that proposed by Otsuka and Ginocchio (1985). The conditions under which the g-boson coupling is eliminated are specified under various approximations and resulting spectra are compared with the exact ones obtained from diagonalisation of an exact s-d-g boson Hamiltonian.

DOI
10.1071/ph880633

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A renormalisation method is developed in which g-boson coupling effects in a transformed Hamiltonian are minimised. The unitary transformation upon which this is based is that proposed by Otsuka and Ginocchio (1985). The conditions under which the g-boson coupling is eliminated are specified under various approximations and resulting spectra are compared with the exact ones obtained from diagonalisation of an exact s-d-g boson Hamiltonian.
We consider a class of Hamiltonians describing a fermion field coupled to a boson field. The interaction kernels are assumed bounded in the fermionic momentum variable and decaying like ∣q∣−p|q|^{-p}∣q∣−p for large boson momenta qqq. A realistic physical case would be p=12p=\frac12p=21​. We impose a spatial cutoff and UV-renormalise the resulting Hamiltonian by subtracting its ground state energy. Our renormalisation procedure works for the physically realistic value p=12p=\frac12p=21​ in spatial dimensions 111 and 222 and for p>34p>\frac34p>43​ in spatial dimension d=3d=3d=3.
The microscopic basis of the Interacting Boson Model for deformed nuclei is discussed. The IBM Hamiltonian is constructed microscopically in the following two steps. In the first step, the collective nucleon pairs of J = 0/sup +/ (S), 2/sup +/ (D), etc. are mapped onto the corresponding bosons. Nucleon-nucleon interactions are also mapped onto boson-boson interactions. This mapping method for deformed nuclei was proposed recently, and it turned out that this method is consistent with the Hartree-Fock-Bogoliubov + angular momentum projection calculation. Low-lying collective states primarily consist of S and D pairs. Consequently, the corresponding boson states mainly consist of s and d bosons, while there are some admixture of g-bosons. In the second step, effects of these g-bosons are included within the s-d boson space by a unitary transformation which transforms a combination of d and g bosons into a new d-boson. By minimizing the coupling between new d and g bosons with an appropriate mixing angle, one can neglect the coupling and obtain the IBM Hamiltonian with s and d bosons. It is demonstrated that the s-d Hamiltonian thus derived indeed reproduces spectra of the original s-d-g Hamiltonian.
To see how closely an interacting-boson-model-like model can be approached, we systematically develop a similarity transformation that unitarizes the simple correspondence version of the Dyson boson mapping of the Ginocchio SO(8) model. For a large region of Hamiltonian parameter space, a one-body plus two-body Hermitian boson Hamiltonian of the interacting-boson-model form is achieved. The similarity transformation consists of a dynamical part and a normalizing part. The dynamical transformation results in a one-body plus two-body Hermitian Hamiltonian and simple transition operators. The normalizing transformation commutes with the dynamically transformed Hamiltonian. This enables the transition rates to be calculated without introducing dual bases. However, the resulting boson model differs from the phenomenological interacting boson model in that transition rates are not calculable from the absolute squares of matrix elements. Dynamically transformed operators are given explicitly, and normalizing transformations for SO(6) and seniority limits are obtained in closed form.