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Ss It of the So Icon Bound-State Model of the Hyperons NUCLEAR PHYSICS A

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Ss It of the So Icon Bound-State Model of the Hyperons NUCLEAR PHYSICS A Nuclear Physics 539 (1990 662-684 NUCLEAR North-Holland PHYSICS A ss it of the so icon bound-state model of the hyperons jërnberg, K. Dannbom and D.O. Riska Department ofPhysics, Universitr Qf Helsinki, 00170 Helsinki 17, Finland N.N. Scoccola Vie Kiels Rohr Institute, 2100 Copenhagen, Dentnark AIltract: We present a simple approximate version of the model for the heavy flavour hyperons, which describes them as bound states of an SU(2) soliton and heavy flavoured mesons, that applies in the limit of a large meson mass. The approximate model leads to an explicit expression for the energy of the bound meson, which reveals its sensitivity to the decay constant of the flavoured meson and also the presence ofa near degeneracy ofthe meson energies in states with equal values of 2j-l+2n_n being the principal quantum number. It also reveals that the contsistcrcy of the model' requires that the decay constant of the heavy flavour meson grow with the square root of the meson mass. The approximate model is finally u~ ::d to predict the magnetic moments of the battom. 'tlyperons. 1 . Introduction The structure ofthe strange hyperons can be fairly well accounted for by describing them as bound states of an SU(2) soliton and strangeness carrying kaons ' -4). A direct extension of this model has also recently been found to provide an at least qualitatively reasonable description of the structure of the charmed and bottom hyperons with resulting spectra that are remarkably close to those obtained by dynamically quite different phenomenological quark models'-"') . Finally it has been shown that if the complete flavour symmetry breaking implied by the differing values of the decay constants in the different flavour generations is taken into account the model leads to predictions for the spectra of both the strange and the charmed hyperons, which are in essentially quantitative agreement with available empirical data and remarkably similar to the values obtained with refined versions of the constituent quark model ") . The bound-state model is a dynamically complete model, which describes the structure of the SU2) soliton and the heavy flavour hyperons on the basis of the same lagrangian model. It is not numerically transparent, however, as the determina- tion of the meson energies requires numerical solutioa of the wave equation for the bound meson states, in addition to the numerical determination of the soliton profile. In the exteHion to the heavy flavour hyperons the very large masses of the D- and -mesons can, however, be exploited to simplify the model . This is because the large meson masses make the meson wave functions very peaked near the centre 0375-9474/92/SO5.00 (K) 1992 - Elsevier Science Publishers B.V. All rights reserved M. Bjbrnberg et al. / Large mass limit 66 3 of the soliton. In this situation the long-range behaviour of the soliton profile becomes unimportant, and it can be replaced by a simple power series expansion near the origin. This simplifies the wave equation to a form that can be solved by the standard methods for harmonic oscillators. A subsequent approximate solution of the resulting quartic equation for the meson energy leads to a simple closed form expression. This energy expression explicitly reveals the strong sensitivity to the decay constant of the flavoured mesons found numerically in ref. "). More impor- tantly it reveals a hitherto unnoticed but very prominent near degeneracy in the bound-meson spectrum of states with equal values of 2j -1+2n, where I and j are the orbital and total angular momentum quantum numbers, and n is the principal quantum number, which indicates the excitation number of the state in each partial wave. The energies of the B-meson states calculated numerically in ref. ') provide a clear demonstration of this degeneracy, which should be common to all versions of this bound-state mode!. Here we shall use the approximate version of the bound-state model to study the sensitivity of the predicted hyperon spectra to the lagrangian density and to calculate the magnetic moments of the bottom hyperons, an application to which it is well suited. In order to study the model dependent- of the spectra we consider, as alternatives to the usual quartic stabilizing (Skyrme) term 1,2,12,13) in the lagrangian density, two versions of a term of sixth order in the derivatives of the soliton 14,15 field ) . This study revealed that although those observables that only depend on the soliton profile - e.g. the nucleon observables - are fairly insensitive to the choice of stabilizing term, the calculated hyperon spectra do depend strongly on the choice. As long as the soliton profile is left unmodified by the meson field component it turns out that the most realistic hyperon spectra are in fact obtained by using the usual quartic stabilizing term. The final application to predict the magnetic moments of the bottom byperons revealed that the predictions obtained with the bound-state version of the topological soliton model are very similar to the results obtained with quark model based approaches. In contrast to the quark model the topological soliton model is, however, dynamically complete - i.e. it leads to predictions for all the magnetic moments in all the flavour generations, whereas in the quark model the magnetic moments of the heavy quarks have to be fitted, or alternatively the heavy quarks have to be assumed to behave as free Dirac particles. This paper is divided into seven sections. In sect. 2 we describe the effective meson interaction as obtained with the Skyrme model, and in sect. 3 we describe the approximate wave equation, and derive the explicit expression for the meson energy. In sect. 4 we analyze the predicted hyperon spectra using the approximate ciGJVd form results. In sect. 5 we discu:;s the consequences of the alternative sixth-order stabilizing terms on thehyperon spectra. In sect. 6 we usethe approximate model to predict the magnetic imoments of the bottom hyperons and in sect. i we give a concluding discussion. 664 M. Bjiirnberg et al. % Large mass limit he e éctive meson interaction 2.1 . THE S®LIT N LAGRANGIAN The bound-state model is based on a lagrangian density for a SU(3)-values soliton field , which is separated into a flavour carrying meson component UM and a SU(2) soliton component U, as`) U =Nf-VM UT~VM . (2.1) ere the soliton field component is defined as u 0) , U;~= (2.2) 0 1 with u being the usual SU(2) hedgehog field if! 1 r 1 ' u = ei: (2 .3) The chiral angle ®(r) is determined by the Euler-Lagrange equations of motion of the lagrangian density specified below. The flavoured meson field has the form i,~_2_ 0 M Um = expi _; , (2.4) . fm L M o where M is one of the isodoublets Du + B ) - (K') D = ( D- B = , (2.5) K° Bo and _fti, the corresponding decay constant. In general the soliton field U should be taken as an SU(5) field in order to encompass all the flavour components, but as long as the flavour carrying component is treated perturbatively to second order the, different flavour sectors can be treated as separate SU(3) flavour groups 5) . We shall first consider the usual Skyrme model lagrangian density ere the operators L,u are defined as L, = U'a,~U. In sect. 5 we shall consider two alternate stabilizing terms of sixth order in the lagrangian density. The lagrangian density has to be augmented by a chiral symmetry breaking term that contains the pion and the heavy meson masses and the corresponding decay constants. For this term we use the form ' 6 ) , ysH= ,' -,(f +2f%,) Tr JU+ U - 2} - ~(J M n7-.Î~m~) Tr{~hx(U+ U)} (2-',_(.i' ~t -.f=) Tr {(1-~1l~)(UL,~L' + L,~LI U' )} . .7) M. Björnberg et al. / Large mass limit 665 Below we shall for simplicity take the pion to be massless, as the pion mass plays no significant role at the heavy mass scales considered. The particular choice of symmetry-breaking term above has recently been shown to be that required for a consistent introduction of the different values of the decay constants of the mesons of different flavour, and in addition it solves the overbinding problem that otherwise plagues the bound-state model "). Finally the model has to be completed by the Wess-Zumino action i1il,. S = - 2 d'x E Tr { L~,L,,LgLßL,,} , (2.8) 240,ff f which contributes to the effective interaction for the flavoured mesons , 2). The Euler-Lagrange equation of motion for the lagrangian density (2.6) leads to the following differential equation for the chiral angle 0(r) [refs. f2 2 op _ 1 ) 1 1 1 . 0"+- , sin 20 -; sine 0 sin 20 -; (0'2 sin 20 + 20" sin' 0) = 0 . e2r r- [r4 r` (2.9) It is convenient to use the dimensionless variable x = 2ef~r. Near the origin the solution to this equation may then be expanded as 0=7r-X+ 2 X3 . (2.10) The coefficient of the linear term (-1) is in fact a numerically calculated number, which is very close to -1 . It should be noted that these coefcie%ts take different numerical values when the pion mass is not neglected. In the limit of a large mass for the flavoured meson this expansion can be used to simplify the effective interac- tion that binds the meson to the soliton. 2.2. THE EFFECTIVE INTERACTION The bound-state model for hyperon structure ;s obtained by expanding the field (eq.
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