Nuclear Physics 539 (1990 662-684 NUCLEAR North-Holland PHYSICS A

ss it of the so icon bound-state model of the

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 , that applies in the limit of a large 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 carrying ' -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 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 ") . 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 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 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 have to be fitted, or alternatively the heavy quarks have to be assumed to behave as free Dirac . 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 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. (2.1)) to second order in the mesonic doublet field (eq. (2.5)) upon insertion in the lagrangian density (eqs . (2.6), (2 .7)) and the Wess-Zumino action (eq. (2.8)). Projection of the soliton on states of good and with a concomitant spin-isospin transmutation of the field leads to a wave equation for the now spin-2 meson field that has the general form a) 2 . a(r)V kn 4 b(r)r- kn -c(r)L2kn -[vo(r)+VIL(r) L]kn

-rnmkn +2wA(r)kn +d(r)conkn =0 . (2.11)

Here (On is the energy of the bound meson. The radial fui,ctions a, b, c, d are defined

666 R. nj6rnberg et al. / Large mass limit

as 4) 1 sine

12 [ sin 20] ~ r r 4ef,, = 1 ,, _ sin' 0 c(r) r2 (2.12c) 4e2 Îimr` ( ® ) ,

~1 P_+, 2 sin2 . d(r)=1+ ® (2.124) 4efM r` The effective central and spin-orbit potentials vo(r) and v,,.(r) are defined as r- 1 O' sin 28 vair) = -i(®" tan 0+ ;®'-) l+ , , --tan20 1+ 2efM 4eÎf 2m r 1 f~ , 2 sin 20 +- 1-; tan 10 0"+-Or- , (2.13a) 2 fM r r` sin`'4 sin'` 2 1 ®,, v®L(r) = 1 + , , _+ r- efM r` 3 sin`' 0 - , , - 0'`'(1-4 sine 150) - 8"sin 8 . (2.13b) 2efMr` [ r,` The function A(r) in the linearly energy-dependent term, which is due to the ess-Zumino action (2.8), has the form 3 sin 2 00' . A(r) - - ,ff r_ (2.l4) 8 f 2 The orthonormality condition for the meson is finally'`)

d'r [(co, +cü") d(r) +2A(r)]k~, k, = S1» (2.15)

In the limit of a very large meson mass mm the bound-state solutions to the wave equation (2 .11) will have very short range. In that limit the radial functions in the wave equation can therefore be replaced by their short-range limits, which are obtained by using the expansion (2.10) for the chiral angle. In this limit the functions a, b, c, d, and A are

a ( r ) = [ 1 +?,_ (1 - 19x2 ) b(r) =?r 1 +X2_, ( 1 - 1sx2) ,

28, c(r) = , d(r)=1+32 45,E`

M. Bj6rnberg et al. / Large mass limit 667

Here the factor X is defined as % =fm/f, The corresponding expansions for the effective interactions vo(r) and vtl,(r) (2.13) are

+ 379 ] vo( r) = 4e2f 2 1 + _ 7 ' (2.17a)) X %2 4 90X2 (1+ ) [1 184 ]J . VIL(r) = 4e2f 4 2 _ + (2.17b) x X 45X In the following section these limiting forms of the interactions will be used to simplify the wave equation (2.11) in order to obtain a version that permits closed form solutions for both the energy and the wave functions.

2.3. THE HYPERQN SPECTRUM

The energy of a hyperon state with isospin I and spin J with n identical mesons in a state with energy w and total angular momentum j is in the bound-state model given as

E=Ms.,+nw+ 1 {(1-c)I(I+1)+CJ(J+1)-c(1-c)j(j+1) } . (2.l8) 212

Here Ms., and .fl are the mass and the moment of inertia of the soliton, respectively. The coefficient c is a hyperfine structure constant, which has the general expression 2-4,6) x c=1-2to dru2 (r) d(r)+(l'-,j~~7~~12j)a(r) fo 1 1 ß(r)+(l-jil Illzj)g(r) - (?.19) + 2j(j-~ 1)

Here the auxiliary functions a, ,B and g are defined as

0(20'+ _sin2 ® a(r) = y(r) -1 d(r) + 2f, sin e" +(1-4sin`28)®'2 , (2.20a) 8e M r r`

3 sine 8 20' 2 ß(r) _ -y(r)+ 3 r2 -sin 0 -+0") -(1-4 sin`' 4®)®' , (2.20b) ße2fM2 ( r

1 sin' g(r) _ . (2.20c) -4e2f.2M r2

The function d(r) is defined above in eq. (2.12) and the function y(r) is given as

[ ®P2 + sin7 ® . y( r) =sin2 10 1 + 1, ,, r (2.2l) e fM J

668 . 8jôrnberg et al. / Large mass limit

e reduced matrix elements in the expression (2.20) are given as 6) -2/(21+1) j=1-2 (2.22a) `17j11TIl121j)= 12/(21+1) j=1+ ;,

(22 1+2)A21+1 j' j-1- = (2.22b) (AjIl Il hj) 121/(21+1) j =1 +; In sect. 4.4 we shall derive a simple but accurate approximate expression for the hyperfine structure constant c. The derivation will rely on the short-range limits of the functions a, 13 and g in the integrand in eq. (2.20), which are a(r) =; -5/2X', (2.23a) ß(r) = -1 +4/X' , (2.23b) g(r) = -1/X` . (2 .23c)

3. The harmonic approximation When the meson mass is large, the solutions of the wave equation (2.11) will be insensitive to the long-range behaviour of the effective interaction and the radial coefficients in the equation. These can then be approximated as in eqs. (2.16) and (2.17) . The resulting approximate wave equation for the reduced wave function u(r) = rk(r) then takes the form (in terms of the dimensionless variable x = 2ef~r) 2 ' )elu 1+ _) u'° -~,'u+ 3e2, (1 - -x")eu+ 1+3, _ +vO(lj) u=0 . X_ Ir-x- X_ x-

ere 1 and j are the orbital and total angular momentum quantum numbers of the meson state and ,u, and E are the meson mass and energy in dimensionless units:

MM oi (3.2) 2e,, 2ef, The two remaining interaction coefficients v_,(lj) and v0(lj) in eq. (3 .1) are defined as 2 I + (3.3a) X-1)_ 5 - 347' -1 184 ) )(1 ) . 1+ ~(~+1)+l(1+l + 9 , (3 .3b) 4 90X~ 4 45% 4 15X- ere Y is defined as the quantum number combination Y =2j-l . (3 .4)

M. Bjbrnberg et al. / Large mass limit 669

In the limit of large meson masses, the meson enegy w is also large and therefore the interaction 50(j) in (3.1) represents only a small correction term. In view of this we have dropped the following term, which is of order r2, in the expansion of the effective interaction vo(r) (2.13a). With the usual Skyrme model parameter values f, = 64.5 MeV and e = 5.45 the harmonic approximation to the linearly energy- dependent term in (3 .1) vanishes at =1.38 fm. The harmonic approximation will therefore be reliable as long as the range of the wave function is smaller than this value. In order to solve the differential equation (3 .1) we employ the ansatz F xz u (r) = xg(x) e-s, , (3.5) which takes into account the long- and short-range behaviour of u(r) explicitly. This leads to the following condition for the exponent S:

- e2 S (3.6) - 27T3(2+1V5 X2) The function g(r) satisfies the differential equation

1 g"+2 1) g-2SvE(2xg'+3g) + X2 g'-2(2zx x 2 2+(1+ 3, E2 - + 3e 2 v0(ij) g = 0 . (3 .7) X X` The general solution to this differential equation, which satisfies the normalizabil- ity requirement (2.15), is a polynomial of the form

n Y' . g(x) = Y_ a2kx2k+ (3 .8) k=0

Here n represents the principal quantum number that indicates the order of the in any given partial wave. Insertion into the differential equation yields recursion relations for the determination of the coefficients a2k and to the following secular equation for the dimensionless meson energy E 2 3e 2(j+ + , E 32 E-(~2+vo(Ij))-2 1+ 2 S,~(3+4n+2Y)=0, (3 .9) X Ir`X - X As in the limit of a large meson mass the state-dependent potential v()(Ij) coefficient represents a minor term in this equation it becomas immediately apparent that the states with the same values of the quantum number combination 2n + Y = 2n + 2j -1 should be near degenerate. This feature is in fact immediately obvious in the numerically calculated B-meson energies in ref. '), when they are plotted as in fig. 1 . Those energy values were obt_ :M,- 2 -.-ith :h.- parameter values f., = 64.5 MeV, e = 5.45 and X = fß/f, = 2. In the following subsection we shall show that the meson limit 67 0 . Bjbraaberg et al. / Large anass E(DeV)

3F512 3D3/2 3Sv2 393,2 293,2 5.0 2D3/2 1 D5,2 2S2 1 F.512

2 P/2 P3/2 1 D 1

1=v2 OF, OD5/2 4 .5 2 - 1P,2 OP/2 OD3,2 OSy2 4 .0 OP/2

I = 0 X =1 X=2 .1=3

Fig. 1 . The energies of the lowest bound B-meson states as obtained in ref. ') with the bound-state model with the Skyrme lagrangian with X = 2.0. The near degeneracy of the levels with equal values of Y+2n, where Y = 2j - l and n is the excitation number, is evident. energy spectrum implied by the approximate equation (3 .9) in fact contains all the qualitative features of the heavy-meson spectra in the bound-state soliton model . The coefficient of x' in the exponent of the wave function (3 .5) provides a natural measure for the viability of the approximate version of the bound-state model. Define x,, as the square root of the inverse of this exponent : (3 .10)

The approximate version of the bound-state model will then provide a good approxi- mation for the matrix elements of such operators for which the expansion in x/x,, converges rapidly. In other words, the smaller x is the better approximation . That expansion in x/x represents a systematic approach to the approximate version of the bound-state model .

eneral features of the meson spectra

4.1 . THE MESON SPECTRA

In order to extract the general features of the bound-state spectra of the heavy flavour mesons that are implied by the energy equation (3 .9) it is convenient to first

M. Bjbrnberg et al. / Large mass limit 67 1

solve the quartic equation (3 .9) approximately . In the limit of a large meson mass the approximate solutions may be obtained iteratively as

E = Ep+71 . (4.1)

Here Ep is the zero-order solution

Ep - (4.2) 1 + 3/X' The first-order correction term 77 has the approximate form

,~ = tio(lj)l2Eo - 3e2/2EO~r2X`+(1+2/X')(3+4n+2 Y)S/ 2 2 2 (4.3) 1 +3/X`+3e /21r X Eo Without reference to any real numerical values these two expressions immediately imply that the energies of the meson states will be ordered in the sequence Y = 0, 1, 2, . . . . The ground state (Y = 0) is thus the P, /2 state. This is followed by the near degenerate S 1 /2-D;/, states, then the P3/2-F5/2 states and so on. This degeneracy is only lifted by the small state-dependent coefficient 0jlj) in the correction term (4.3), which makes the energies of the j = I + ; levels slightly higher than those of the corresponding j =1- 1 levels. This structure is very evident in the calculated B-meson spectra obtained with the bound-state model in ref. 6), and which are shown in fig. 1 . The energy differences between the near degenerate levels in this example are typically less than 40 MeV, which is small in comparison to the excitation energies, which are of the order of 400 MeV.

4.2. THE LARGE MESON MASS LIMIT

The accuracy of the approximate energy expression (4.1) grows with meson mass. It is therefore natural to consider the limiting expression that is obtained in the case of a meson mass that is much larger than the depth of the effective binding potential: m (4.4) 1 -~- 3/X-' . From this result it may be inferred that this model will lead to a serious overbinding for the heavy flavour hyperons if the decay constants are taken to be equal in all flavour generations (i.e .f~ =fM) as was originally done' -6). The corresponding value m would in the case of the D-meson represent a binding energy of only 934 MeV for the meson ground state, which is about 500 MeV too small . If instead, as was done in ref. 6), one employs the ratio fo/f, =1 .8, which falls within the empirical range 1 .7 :L 0.2 [ref. '9)], the meson energy increases to 1347 MeV, which is essentially what is required in order to obtain the empirical value for the mass of the Ac- hyperon (2285 MeV) [ref. ")].

672 M. Bjdirnberg et al. / Large mass limit The remarkable accuracy of the limiting expression (4.1) for the case of the -meson (m" =1869 MeV) and the charmed hyperons should be noted : the numeri- cally calculated D-meson energies are in the case off, =fn 760 MeV and in the case off =1 .8f., 1301 MeV both of which are within 20% of the values obtained from the expression (4.1) above (the larger value is in fact off by no more than 3.5%!). The reason for this is the fact that the terms in the numerator of the correction term -q (4.3) tend to cancel in the case when m = mj, . In the case of the B-mesons (mB = 5279 MeV) the large mass limit of the energy expression (4.1) gives the meson energy 2639 MeV in the limit fB =f.,, whereas with ./B= 2.0f., one obtains the value 3990 MeV. These should be compared to the corresponding numerically calculated values in refs. '),6), which are 2390 and 3940 MeV, respectively . Again the approximate expression gives a value that is within 10% of the correct one. These numbers do in any case very clearly reveal the importance of taking into account the fact that the decay constant of the heavy flavour mesons are considerably larger than the pion decay constant. They also demonstrate very explicitly the finding in ref. 6 ) that the spectrum of the B-flavour hyperons cannot be realistically described with the conventional Skyrme model parameters, unless the ratiofB/f, is increased beyond the present (though uncertain) empirically obtained range fB/.Î:~ =1 .1-1 .6 [ref. ")] to a value of the order of 2.5-2 .6 . Such a large value would on the other hand lead to a probably unrealistically large hyperfine splitting of the B = -1 hyperon states.

4.3. THE EXCITATION SPECTRUM The separations between the lowest levels in the B-meson spectrum calculated in ref. 6) and shown in fig. 1 are roughly equal. The approximate energy expression (4.2) contains this feature - at least qualitatively - for the lower lying excited states. The harmonic approximation to the binding interaction should in fact not be expected to describe the highest excited states very accurately, as the corresponding wave functions are sensitive to the long-range behaviour of the interaction, which cannot be described realistically in the harmonic approximation. From eq. (4.3) we may derive the following approximate expression for the level spacing ® for states with definite Y = 2j - l:

8ef,(1+2/X 2 ) S )V (4.5) (1+3/ `+3e2/272X2EO) NrEO In the case of the B-meson mass of 5279 MeV, and with fB = 2f,(X = 2) this gives the value 420 MeV for the excitation energy. This agrees well with the typical excitation energies 400-490 MeV for the B-meson states found in ref. 6), and shown in fig. 1 . The approximate expression (4.5) will, however, give too large values for the decreasing level spacings between the higher excited states, a description of which would require going beyond the present harmonic approximation.

M. Bjbrnberg et al. / Large mass limit 673

4.4. THE HYPERFINE CONSTANT

In the limit of a large meson mass the short-range limit of the integrand in the expression (2.19) for the hyperfine structure constant c will yield a sufficiently good approximation. Since the integrand apart from the meson wave function in that case is constant, the remaining wave function integral is given by the normalization condition (2.15), once the corresponding constant short-range limit of the function A (r) (2.14) is used in the normalization integral. One then obtains the following explicit expression for the hyperfine structure constant c:

do+(I jJITIJI21)ao+ [2j(j+l)]- 'ßo+(hjllLlihj)go c=1- (4.6) do+Ao/w

Here ao, (3o and go are the short-range limits (2.23) of the corresponding functions in the integrand (2.19) and do, and A  the short-range limits of the functions d(r) and A (r) given in eq. (2.16). In the case of the ground state (I =' , j =;) the expression (4.6) reduces to the very simple form

(4.7)

This expression shows that c increases with the ratio X =fM/f r and decreases with increasing meson energy or - equivalently - meson mass. The expression (4.7) for the hyperfine structure constants provides an excellent approximation to the corresponding numerically obtaine-. values in the case of the bottom hyperons. Consider again the example with the typical soliton parameter values f, = 64.5 MeV and e = 5.45 [ref. ")], and with X =fß/f, = 2.0. With these values the approximate B-meson energy given by eq. (4.4) is 3990 MeV (sect. 4.2 above), and thus according to eq. (4.7) the hyperfine structure constant c is 0.15 . These values are very close to the corresponding numerically obtained values 3940 MeV and 0.18 . This example, beyond demonstrating the accuracy of the expression (4.7), clearly demonstrates the crucial role of the ratio X =fm /f, in the bound-state model. The expression (4.7) also shows that one cannot increase X far beyond the value 2. which otherwise would be tempting as it would lead to better predictions for the mass of the AB hyperon. The consequence of increasingX is a corresponding increase of c, which would imply a much larger XH - A H splitting than what is expected on the basis of quark model considerations. The general expression (4.6) shows that the hyperfine structure constant will vanish in the limit of a very large meson mass in the higher partial waves. In the ground state the limiting value for c depends on the value of the decay constant ratio X, but as long as X is not far from the value VN, c will also become very small for the ground state in this limit.

674 119. Bj6rnberg et al. / Large mass limit

e. st i izi g ter o sixth order

5.1 . THE LAGILANGIAN DENSITY

The approximate methods developed above can be used to study the consequences on the predicted hyperon spectra of using a term of sixth order in the derivatives in place of the usual quartic Skyrme term to stabilize the soliton solutions in the lagrangian (2.6). At the level of SU(2) this sixth-order term can be written in the form y6 = -2s6B,,,,B" , where is the anomalous current

B' = 1 2 sTr L,,L,,Lß . 5.2 247r This sixth-order term can be viewed as the limiting form of the coupling to an omega meson field that obtains when the omega meson mass grows beyond bounds 14). When the field operator U belongs to a larger unitary group than SU(2) the form of the sixth-order term (5.1) is not unique. An alternate form, that reduces to the usual one in the case of SU(2), would be y6 = -E6 Tr b,,,b" , (53) where bl is the operator 1 bl = 2 e"""LL"Lß . (5 .4) 127r In the case of SU(2) the current operator b" reduces to the usual baryon current. The predictions for the low-energy observables of the SU(2) - i.e. the and nucleon resonances - are rather insensitive to the choice of stabilizing term in the soliton lagrangian - i.e. whether or not the sixth-order term (5.1) is used in place of the quartic term - as long as the parameters in each case are chosen so '4). that the empirical values of the nucleon and A-',:. mamass vali!es are reproduced e shall here show that in complete contrast to this the predictions for the hyperon structure that are obtained with the simplest sixth-order term Y6 (5.1) are quite different from those obtained with original Skyrme lagrangian (2 .6) in the bound- state approach . The presence of a sixth-order term of the form (5 .1) or (5 .3) adds a term ofthe form _E6 (sin4 e 0,, sin2 ® sin 2® ®,, - 2 sin4 ® ®, 4 4 4 5 (5.5) Ir r r r to the l .h.s. of the differential equation (2.9) for ® [ref. 'g)] . In this case the natural dimensionless variable x should be defined as x = 7r 2f_~/ s6r. The expansion for 0

M. Björnberg et al. / Large mass limit 675 near the origin becomes

0=1T -X-sx3 (5.6) Here the coefficient -1 of the linear term represents an approximation to a numeri- cally obtained value. When the sixth-order term (5.1) is used in the soliton lagrangian in place of the quartic term the parameter values that lead to the empirical values for the nuclear and 633 masses are f, =76 MeV and E6 =1.08 fm.

5.2. THE EFFECTIVE INTERACTION Although the two versions (5 .1) and (5.3) of the stabilizing terms of sixth order lead to the same results for the SÜ(2) soliton, they lead to formally quite distinct forms for the effective interaction in the meson wave equation (2.12), and also to very different predictions for the heavy-flavour hyperon spectra. Consider first the simpler version Y6 defined in eq. (5 .1). This contributes a term of the form

EV62 sin2 0 2 L(r) = Ir4f2 r4 ®" -r 0' sin 8 + 2 cos ee'21 (5.7) M [( to the spin-orbit interaction (2.13b), but gives no contribution at all to the remaining radial functions in the wave equation (2 .11). The fact that Y6 does not contribute to the coefficient d(r), which consequently takes the simple value 1, implies that the heavy flavour hyperon energies that will be predicted with this stabilizing term will be much larger than the corresponding predictions obtained with the Skyrme model. This follows from the approximate energy expression (4.4), which when using the term y6 would become w = m. The meson energy would then be very close, if somewhat smaller than, the mass of the flavoured meson. This is in fact borne out by numerical calculation: employment of the sixth-order term Y6 with the parameters quoted above in sect. 5.1 gives the energies of the bound D- and B-mesons in their ground states as 1340 and 4690 MeV, respectively, whereas with the Skyrme model the corresponding energies are only 769 and 2390 MeV, respectively (in obtaining these numbers fB was set equal to f,) . In the following subsection we shall show, that although the large meson energies obtained with the simple sixth-order term may appear as an improvement over the Skyrme model - especially in thecase of the B-mesons - this version ofthe sixth-order term on the other hand leads to very poor predictions for the hyperfine structure constant c. In contrast to the sixth-order term _T6 (eq. (5.1)) the alternate version Y6 (eq . (5.3)) contributes to all the radial functions in the meson wave equation (2.12) and to both components of the effective interaction (2 .13). The contributions to the

676 M1. Bj6r®iherg et al. / Large mass limit

coe cients - are , 6 sin' 0 (5.8a) ®. . " r c(' sin' 0 (2 sin' ® , , , , (r) _ a - ®sin 0(4-10 sin' ;®) - 0 tan ~0 sin` 0 (5 .8u) 9~rfM r L r 2 E sin'' 0 c(r) _ %11 ,~ 0',`, (5.8c) 9 7r r

E :! sin2 0 !îLW~ 9 ( (5 .8d) "° 2 ' ) The contributions to the central and spin-orbit interactions are correspondingly 2 E sin'` 0  2 P , 20 v(,(r)= î.4 ® -_0 sin 0+(3cos®-1)0`+ _;0 `,(1+2tan `'_50), ssin 187r .Î n7 2 , + 0"tan ;O sing 0 - 0' tan ;0 - sin` 0 - ®'sin 0(4 -10 sin21 0) (5.9a) Ir il, 20 =- E6 , sin ~'~L(r) ® -?®') sin0+(3cos0-1)0''-®''sin0tan ;0 187r fM r 1( r (5 .9b) Below we shall discuss the consequences on the calculated hyperon spectra of the different effective meson-hyperon interactions that are obtained with the alternate sixth-order terms Y,' and

S.3. THE HYPERFINE STRUCTURE

The most important difference between the two alternate forms of the sixth-order stabilizing terms Y" and Y" is that the version Y6 leads to far larger meson energies, as in the limit of large meson masses the corresponding approximate energy expression is

W=M (5.10)

as noted above. When using the sixth-order stabilizing term Yb the approximate energy expression that corresponds to the Skyrme model expression (4.4) is on the other hand

tn (5.11) 1+4/3X-- ~ This can be derived by the same method as used in the case of the Skyrme model in sect. 4.2 by using the short-range limit of the function d (r) defined in eq . (5.8d) .

M. Björnberg et al. / Large mass limit 677 This is fairly sirr,ilar to the Skyrme model result (4.4), although it implies somewhat larger energy values. Hence the sixth-order term Y6 leads to results that fall in between those obtained with the original quartic (Skyrme) term and those that are obtained with the simpler sixth-order term Y' (5.1). The contribution to the hyperfine structure constant c in eq. (2.20) from the _T6 sixth-order term is

00 E w sin2 e [2e,2 e" cb 4 dr cos e + sin e](t2 ' z !'- ') = - 2~r fMZ fo u2(r) r 1 JII II2~ 1 0)8'2 [2(1-cos -8"sin 0] (5.12) +2J(;+1) The reduced matrix element of the spin operator z is defined in eq. (2.24a). In the limit of a large meson mass the integrand in the expression (5.12) can be approximated by its constant short-range limit. Combination with the contributions from the quadratic term in the lagrangian density given in eq. (4.8), then leads to the following simple expression for the hyperfine structure constant as obtained with the sixth-order term ~6 : ) d _ _ 1+-1(1-4/X2)[(Il II,7II11j - 1/J(J+ 1)] 3 (5 .13) cb-1 1+3a /87 (0f22, In the case of the ground state this finally simplifies to 4 c6=1- 2 (5=1 ~) X+(3Tr/faE6W) .%/2E6 From this expression it can be concluded that the simple sixth-order term leads to very poor predictions for the hyperfine splittings of the heavy flavour hyperons. Consider as an example the case of the B = -1 hyperons. In this case, with the parameter choices E6 =1 .08 fm, a = 2.69 fm - ' and fr = 76 MeV (with f, =fm) the energy of the B = -1 meson in the ground state is 4690 MeV. With these values a non-negative value for c can be obtained only by choosing fm/f" to be larger than 2, which is above the empirically determined range 1 .1-1 .6 [ref.' )]. With fm =f, the expression (5.13) gives the value -2.5, which is close to the corresponding numerically obtained value -2.7. The conclusion must therefore be that although the simple y6 term leads to larger and in some ways better values for the meson energies than the usual quartic (Skyrme) term ., the poor prediction of the hyperfine constants above rules out this term unless it appears in a linear combination with the quartic term. The alternate sixth-order term ~6 ;5.3 ; leads to values r the hyperfine structure constant c that falls between those that are obtained with the quartic term and those obtained with the sixth-order term `e6 . The contribution to the hyperfine constant

6'® M. Björnberg et al. / Large mass limit c is in the case of the sixth-order term Y,' a , i b a 2S6 ~ sin2 A0 c6 =c6 - dru (r) , 9 f w o r` 0)0,[sin3 0_0" x (1 + cos sin 0 cos 0 + 6'`(1- 3 cos`' 0 ) r

+2®''' cos 0 sin2 8 1 (1!2jllTlll-j)

1 sin3 8 0)e'2 (l+cos ®) 0"sin 0 cos 8-30' -sin2 0(1+5 cos + 2'(jj,+1) [ r I - sin 8 (I~jI +4c0S2 10 cos 0 e'-+ 1 L 11 I;j) r

Here c6 is thecontribution (5.11) to the hyperfine structure constant from the simpler sixth-order term Y6 (5.1) . 1n the limit of a large meson mass this simplifies to

_ _ 1+ ;(1 20/9X2)[(jz.lIIz61Aj) - l/j(j+l)] - (819X2 )( 12jJ)L1) 11J) c61 2 do+(3 ~/f~rc6wX ) fir/«6 (5 .l6)

where do is the short-range limit of the function (5 .8d) :

(5do == 1 +4/3X2 , .l7) which also appears in the approximate energy expression (5.10). Comparison of the form of this expression to the corresponting expression (5.13) that is obtained with the simpler sixth order term y6 shows that the hyperfine structure constant given by the expression (5.16) is much larger than that obtained with the expression (5.13). The reason is that the denominator in the fraction in eq. (5 .16) is srr4aller, and the denominator larger than in the fraction in eq. (5.13) . Thus the value of the hyperfine structure constant c6 will lie between the value of the hyperfine constant c and that obtained with the usual quartic stabilizing term (4.6).

6. The magnetic moments

6.1. APPROXIMATE EXPRESSIONS FOR THE MAGNETIC MOMENTS As a final application of the approximate version of the bound-state model we consider the magnetic moments of the heavy flavour hyperons. The magnetic moments ofthe hyperons are formed as the sum of a soliton and a meson component. As the mesonic contributions to the magnetic moments are approximately propor- tional to the square of the r.m.s. radius of the meson wave function, it is obvious

M. Bjbrnberg et al. / Large mass limit 679 that the relative significance of these contributions will become unimportant as the meson mass grows and the spatial extent of the wave function (3.5) correspondingly shrinks. Below we shall show explicitly that already in the case of the bottom hyperons the mesonic contribution only represents a very small correction term so that in effect the magnetic moments of the bottom hyperons are dominated by the soliton contribution. The complete expressions for the magnetic moments of the strange and charmed hyperons have recently been given in ref. 1y). By a slight modification these can be generalized to apply to the case of the bottom hyperons as well. Accordingly we write the magnetic moments as

lu = lus+luv , (6.1a) lus = AI"'' +(cA- QB)J3 , (6.1b) 1 . liv =-2(C+IQI D)y(I"')I3'J3 (6.1c) Here Q is the flavour quantum number, that describes the hyperon state. The coefficivents y(IS°') are numerical coefficients, which depend on the quantum numbers of the soliton. For I'°'=0, 2 and 1 the values of y(IS°') are 0, 3 and 1, respectively 2°). The coefficient c is the hyperfine structure coefficient defined in sect. 4.4. The coefficients A and C are given by the nucleon mass M N and the moment of inertia ,fl and mean square radius of the soliton as

A = (r2)ß , (6.2a) 3 C = 2M NI2 . (6.2b) Finally B and D are radial matrix elements of the meson wave functions, which are defined as 1 B = MN d3r k2 cos2 20+ 4e2f2 f I M 2 20'2 x 4 k2 sin2 0 COS2 2'0+k COS2 20 + 3kk'0' sin 0 (6.3a) r 11 ,

D = 3MN 1 d3r i k2 cos2 20(1-4 sin2 20)

1 k2 ~ , 2 1 2 OP-2 - 1 +4 , [4 2 sin2 0 cos" 20(3 - S sin` 20) + k cos~ 20(1-18 sin` 20) e2.ÎM r

+2k''` sin2 0 + 3kk'0' sin 0(3 -4 sin`' 20) + m22 d3 r 0'k2 sin`' 9 . (6 .3b) Ô?l fM These expressions contain all the contributions to the magnetic moments of the meson that arise from the effective interaction in (2.12) . Parts of these expressions have been given previously in refs. 2°-22) .

680 M. Björnberg et al. / Large mass limit Consider now the short-range approximation (2.10) in the integrands, with the corresponding approximate wave functions (3 .6), (3.8). In the case of the lowest lying states (`e =0) it then is obvious that the matrix elements B and D are proportional to the mean square radius (r2) of the meson wave functions. For the approximate wave functions (r2) is obtained as 3/4S./w- times the integral of the square of the meson wave function, which in turn is given by the normalization condition (2.15). Using this fact, and using the expression (3.6) for the range parameter S, one finally obtains the simple approximations

+;X2 + sX 2X ( .4a) w l + ( e3/ ~2)(fa/ w) 24

MN 1 _ hr D _ ~ 2 2 1 16K (1 +37X2) 1 +ZXZ 6w 3X (e VAO) 2 _ MNK e 1 +;X 1+1X2+ (6.4b) 32f2 (e3/,f2)(f~l w)' where K is the numerical constant (6tc =41r 3/5e . .5) These expressions already give very good approximate values in the case of the charmed hyperons. With the standard Skyrme model parameters (f = 64.5 MeV, e = 5.45), and with X =1 .8 the D-meson energy is 1301 MeV (sect. 4.2) . From eq. (6.4) one then obtains the value B = 0.37 and D = -0.25. The value for B is close to the numerically obtained value 0.29 in ref. ") but the value for D is larger than the numerically obtained value -0.10. The reason for the poorer prediction for the coefficient D is the cancellation between two large terms in eq. (6.4b), which emphasizes the inaccuracy of the approximate model. As the accuracy of the approximate model that underlies the expressions (6.4) grows with meson mass, the nevertheless reasonably successful application in the case of the charmed hyperons implies that the expressions should be very reliable in the case of the bottom hyperons, and we therefore below use them to predict the magnetic moments of the stable B-hyperons.

6.2. THE MAGNETIC MOMENTS OF THE BOTTOM HYPÉRONS To calculate the mesonic contribution of the bottom hyperons using the approxi- mate expressions (6.4) requires knowledge of the meson energy w and the ratio X in addition to the Skyrme model parameters f and e. In this case the dependence ofthe coefficient B and D on both V and w is small, and furthermore both coefficients are small in comparison with the coefficients A and C (6.2), which give the soliton contribution. Therefore even a crude model for the 13-meson wave functions should suffice. We shall therefore use the usual Skyrme model parameters as above, and in addition take X =1 .8, which is only slightly above the empirical range 1.1-1 .6

M. Bjbrnberg et al. / Large mass limit 681

TABLE 1 The magnetic moments ofthe bottom hyperons as functions of the coefficients A, B, C and D (eqs. (6.2) and (6.3)). The coefficient B is defined as B = cA- B

lu( A b) = 2B Wybb) = - 6A+3B -9(C+2D) N-(lb)=2A 68+2(C+D) u( - bh)=-6+2h+-9(C+2D) u(10 ) = 3A - 6à p(e**t°»=2A+B+3(C+2D) W(X- ) = 2A-6B-2(CtD) IL(,r*,h)=2A+B-3(C+2D) ~(Ib+) =A+2B+(C+D) A(ftibb)=3B w(X*°)_ =A +2B N(Zb ) =A+26-(C+D)

[ref. ")]. With these parameter values co = 3730 MeV [ref. ")]. The values for the coefficients B and D given by the expressions (6.4) are then -0.14 and -0.02, respectively. It is worth noting that this value for D is almost totally due to the Wess-Zumino term in D, which was neglected in refs. 2°,22). For the coefficients A and C that describe the soliton contribution one obtains the values 0.555 and 2.401 with the same Skyrme model parameters as above. Since these do not lead to the empirical values for the magnetic moments of the nucleons, we expect that more reliable predictions will be obtained by using the values A = 0.88 and C = 3.53, which do lead to the empirical nuclear magnetic moments. In table 1 we list the expressions for the magnetic moments of the stable pure 13-flavour hyperons in terms of the coefficients A, B, C and D. The corresponding numerical values are listed in table 2. In that table we also list the quark and bag model predictions given in ref. 25) . The very small predicted value for the 11 h , which is entirely due to the 13-meson component of the bound meson-soliton system reveals the shrinking of the mesonic contribution with growing meson mass.

TABLE 2 The predicted values of the magnetic moments of some pure B-flavour hyperons as obtained with the bound-state model using the approximate magnetic moment coefficient (approx.) obtained from eq. (6.4) and using the numerical obtained coefficients (numerical). The quark and bag model predictions are from ref. 25). All moments are given relative to the magnetic moment

Bound-state model Quark Bag model model approx. numerical

n h -0.001 0.003 fitted fitted h 1 .01 1 .03 0.92 0.83 0.20 0.20 0.24 0.21 -Yh -0. -0.45 -0.40 0 hh -0.31 -0.32 -0.25 -0.22 bb 0.21 0.23 0.08 0.05

682 M. 8j6rnberg et al. / Large mass limit The predicted values of the magnetic moments in table 2 provides yet another example of the fact that the results obtained with the topological soliton model tend to be very similar to those that are obtained on the dynamically quite dissimilar 5,`' quark model based approaches ). A qualitative explanation of this by nowgeneral feature in terms of an induced gauge structure associated with the fast and slow degrees of freedom that correspond to the massive and light flavour quarks that form the baryons is given in ref. '4).

7. Discussion Above we have shown that the simplified approximate version of the bound-state model for the heavy flavour hyperons gives an accurate description of the bottom hyperons and already a fairly good description of the charmed hyperons. The virtues of the simplified model is that it yields closed form expressions for both the meson energy and the corresponding bound-state wave functions which can be used to calculate the hyperfine structure constants and the magnetic moments. The closed form expressions reveal the dependence of the observables on the key parameters in the model, as for example the decay constant ratios fu/f, and f,/f,. Thus it provides an explicit explanation for the observation in ref. ") that unless the ratio f/f, is allowed to take its empirical value the model cannot provide a quantitatively accurate description of the charmed hyperons. The approximate version of the bound-state model was then used to study the sensitivity of the predicted hyperon structure to the model used for the lagrangian density. It was shown that while the nucleon observables are fairly insensitive to the choic° of stabilizing term in the lagrangian density, the hyperon spectra depend strongly on whether the usual quartic (Skyrme) term, or a term of sixth order in the derivative of the soliton field is employed. The approximate expressions (4.4), (5 .10) and (5.11) for the meson energy, which apply for the different stabilizing terms, do, when combined with the requirement that the mass of the lowest hyperon state in each flavour generation should exceed the corresponding meson mass, imply that the decay constant ratio X ==fm/f, must grow with the square root of the meson mass, at least in these simplest versions of the bound-state model. This can be seen from the hyperon mass formula

MH - MOI + (0 -+- Wh.f. ) 7.1) where MH is the hyperon and M,., the soliton mass, and Ohf. the hyperfine energy, which as shown in subsects. 4.4 and 5 .3 should vanish in the limit of a very large meson mass m. The approximate energy expressions (4.4), (5.10) and (5.11) can be written in the general form

W = m/C(X) , (7.2) where the factor C(X) equals 1-+-3/X` in the case of the quartic, 1 in the case of

M. Bjbrnberg et al. / Large mass limit 683

the simple sixth-order stabilizing term Y6 (5.1) and 1 +4/3X`' in the case of the sixth-order stabilizing term y6 (5.3). The requirement that MH - m be positive combined with the hyperon mass expression (7.1) in the large mass limit then leads to the inequality M"oj > 1- 1 . (7.3) m C(X) When using the simple sixth-order term °L6 (5 .3) this inequality is automatically satisfied as C =1, and thus this stabilixing term does not imply any restriction on decay constant ratio X. On the other hand, in the case of the quartic (Skyrme) stabilizing term the inequality (7.3) implies that the decay constant ratio should be bounded from below: X > 3m/2M,., . (7.4) In the case of the D-meson, with M,., = 866 MeV this inequality would imply that X > 1 .79, and thus with the value X =1.8 used e.g. in refs. .6,11), the positivity require- ment is satisfied. In the case of the B-meson the inequality (7.4) would, however, require that X exceed 2.3, a value which is clearly above of the empirical range 1 .1 .-1 .6 [ref. ")]. This shows that with this stabilizing term alone, the bound state model cannot provide a completely quantitative and consistent description of the bottom hyperons. The alternate sixth-order stabilizing term (5.3) relaxes the bound (7.4) to X> 2m/3Mso, . (7.5) For the B-meson this requirement implies that % > 1 .53 a value, which does fall within the empirical uncertainty range. The inequalities (7.4) and (7.5) do however show that the bound-state model, with the exception of the special case of the simple sixth-order stabilizing term (5.1), requires that the decay constant ratio fm/f, grow as the square root of the mass of the flavoured meson. This is similar to the Van '6). Royen-Weisskopf condition in the quark model Very large values for % do however have the undesirable consequence of driving the hyperfine structure con- stants (4.7) and (5 .16) negative or making them unrealistically large (5.14). The final application of the approximate version of the bound-state model was to obtain explicit predictions for the magnetic moments of the bottom hyperons. The value of using the approximate version here is that it reveals the insensitivity of the predicted bottom hyperon magnetic moments to the details of the mesonic contribution . The small spatial extent of the B-meson wave functions has the consequence that the mesonic contributions are small in comparison to the soliton contribution . This situation provides an example of the general feature of SU(2) ~~ 2a symmetry restoration in heavy quark systems `3 ' ) . The analysis presented here of the bound-state model for the hyperons applies only to its simplest version in which the soliton is treated as inert and no explicit

684 M. Bj6rnberg et al. / Large mass limit

fields are included in the description. It is natural to expect that the model would be considerably improved by explicit inclusion of the heavy-flavour vector meson fields, as the strange and charmed vector mesons have masses that lie only a few hundred MeV above the masses of the corresponding pseudoscalar mesons 5,6). Inclusion of the heavy flavour vector meson fields will most certainly be required for a completely satisfactory description of the bottom hyperons for which the simplest versions of the model cannot simultaneously predict the ground- state energy and the hyperfine splittings in a quantitatively satisfactory way.

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