On the Renormalization of the IBA by the g Boson

Abstract

The IBA-2 proton-neutron Hamiltonian is given by^l–2

$${H_{IBA2}} = \varepsilon \left( {{n_{{d_\pi }}} + {n_{{d_v}}}} \right) + \kappa {Q_\pi } \cdot {Q_v} + {M_{\pi \nu }} + {V_{\pi \pi }} + {V_{vv}}$$

(1)

where κ < 0

$$\begin{gathered} {Q_{\pi \left( v \right)}} = \left( {{d^\dag }s + {s^\dag }\tilde d} \right)_{\pi \left( v \right)}^{\left( 2 \right)} + {\chi _{\pi \left( v \right)}}\left( {{d^\dag }\tilde d} \right)_{\pi \left( v \right)}^{\left( 2 \right)} \hfill \\ {V_{\pi \pi \left( {vv} \right)}} = \sum\limits_{L = 0,2,4} {\frac{1}{2}} {\left( {2L + 1} \right)^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}}C_L^{\pi \left( v \right)}\left[ {\left( {{d^\dag }{d^\dag }} \right)_{\pi \left( v \right)}^{\left( L \right)}x\left( {\tilde d\tilde d} \right)_{\pi \left( v \right)}^{\left( L \right)}} \right]_0^0 \hfill \\ \end{gathered} $$

(2)

and M_πv is the Majorana operator which separates the totally symmetric proton-neutron states from states of mixed symmetry. This Hamiltonian has been used by several authors^1–4 to do phenomenological analyses of nuclei in the mass region Z = 50−82 and N = 50 to N ≥ 126, in which the parameters ε, κ, χ_π, χ_v (and perhaps the C_L’s) are determined by fitting the low-lying energy spectra for an individual nucleus. One would like to obtain values for these parameters which vary smoothly with neutron (or proton) number and then to understand the physical origins of these parameters. Another interesting question is how these parameters are affected by terms left out of the model Hamiltonian, such as the g boson, i.e. proton or neutron pairs coupled to J = 4. One approach to studying this problem is to expand the symmetry group of the IBA from SU(6) to SU(15) in order to include the degrees of freedom associated with a g boson. One can avoid this difficult task by including the effects of a g boson on the s-d boson model space through second order perturbation theory.

Research supported in part by NSF Grant No. PHY-7902654.