Strange baryons in a chiral quark-meson model: (I). SU(3)-symmetric model

Nuclear Physics A506 (1990) 367-391

North-Holland

STRANGE BARYONS IN A CHIRAL QUARK-MESON MODEL

(I). SU(3)-symmetric model

Judith A. MCGOVERN and Michael C. BIRSE

Department of Theoretical Physics, University of Manchesrer, Manchester, Ml3 9PL, UK

Received 2 August 1989

Abstract: The chiral quark-meson model based on a linear a-model is extended to three flavours of

quarks. We describe the choice of parameters and the features of the hedgehog solitons both in

the SU(3)-symmetric limit and with realistic symmetry breaking. For the case of unbroken symmetry the approximate projection method of Birse and Banerjee is used to calculate properties of the

octet baryons. The magnetic moments are found to be in reasonable agreement with experiment,

with most of the discrepancy being accounted for by the fact that the kaons and pions are degenerate.

Lowest-order cranking is used to calculate the mass splittings between the baryon multiplets. The

isospin cranking which governs the octet-decuplet splitting is very similar to that in the SU(2)

model. The moment of inertia for “strange” rotations gives the energies of states which must involve

mesonic excitations. It is found to be small, indicating that such states, if they exist, lie well above

the nucleon.

1. Introduction

QCD is now almost universally believed to be the underlying theory of strong

interactions ‘). It is based on the local gauge group of colour, SU(3&, and fermions

(quarks) which transform as the fundamental representation of this group. Where

quarks occur in iV flavours, the flavour group SU(N) commutes with the colour

group, and the lagrangian for massless quarks is invariant under the larger chiral

group SU( N)L x SU( N)R. This theory is consistent with the experimental evidence

for three colours and does not disrupt the weak selection rules of the SU(2) x U( 1)

theory of weak interactions. It also possesses asymptotic freedom which accounts

for the observed scaling behaviour of strong interactions at high energies.

However, other observed properties of hadrons at low energies are dominated by

the long-distance, strong-coupling regime of QCD, where perturbation theory breaks

down. The most obvious of these properties is confinement; no free quarks or any

other coloured objects have been seen. Another is dynamical symmetry breaking in

which the chiral SU(N) x SU( N) flavour group is spontaneously broken to SU( N)

so that the vacuum is no longer invariant under axial transformations. As a result

of the Goldstone theorem this produces N* - 1 massless pseudoscalar bosons which

are identified with the lightest mesons - pions for N = 2. Explicit symmetry breaking

terms such as quark mass terms in the underlying lagrangian would give the

0375-9474/90/$03.50 0 Elsevier Science Publishers B.V. (North-Holland)

January 1990

368 J.A. McGovern, M.C. Birse / Strange baryons (I)

pseudoscalars mass and lift the SU( N) degeneracy. The absence of parity doubling

of the hadrons, and the success of the PCAC theories which incorporate dynamical

symmetry breaking, are strong evidence that any theory of strong interactions must

also be in this phase at low energy.

One possible way of treating the non-perturbative region of QCD is to regularise

the theory by putting it on a lattice of discrete space and time points. Confinement

and dynamical symmetry breaking do occur in such calculations but detailed

descriptions of hadronic structure and interactions obtained in this way will probably

have to await a new generation of computers.

An alternative approach, and the one that is followed in this paper, is to use

phenomenological models which, it is hoped, capture the essential features of QCD

in the low energy region while being easier to handle. Ideally such a model could

be shown to be equivalent to QCD, but while considerable progress has been made

recently in exploring the connections between QCD and various models, and between

the models themselves, a rigorous derivation is still lacking ‘).

One such model which stresses the role of chiral symmetry and symmetry breaking

is the linear o-model whose solutions for two flavours have been explored by Birse

and Banerjee ‘) and by Kahana, Ripka and Soni “). One of the attractions of this

model is that it incorporates PCAC and the results of current algebra 5).

Unlike the similar Skyrme model 6), the model involves explicit quark degrees of

freedom, allowing one to calculate non-local effects due to virtual quarks in the

effective action. The Skyrme model relies on an expansion of the effective action

as a series of local terms involving powers of derivatives of the fields. The convergence

of such an expansion is questionable since the range of the non-localities is given

by the inverse of the dynamical quark mass, and this is expected to be comparable

to the nucleon radius. Further, attempts to derive the coefficients in the derivative

expansion from more fundamental models do not lead to stable solitons ‘). AS well

as avoiding some of the problems encountered in trying to approximate effective

theories for QCD by local lagrangians involving only mesons, the use of explicit

quarks means that this model is not tied to approximation techniques which rely

on the large-N, limit. We are thus able to use projection methods which take into

account the presence of N, = 3 valence quarks ‘).

Since quarks and current algebra were both introduced to account for the grouping

of hadrons in SU(3) multiplets, it is natural to try to extend the model to include

strangeness. In the present paper we explore the properties of the three-flavour

quark-meson model, concentrating on the SU(3)-symmetric version. Baryon proper-

ties are calculated using an extension of the approximate projection technique of

ref. 3), in which spin and flavour quantum numbers are assumed to be carried by

the quarks alone. We also calculate the spectrum of excited baryons using the

semiclassical approach of cranking 8*9). Brief accounts of our results have been given

previously “,l’). In many repects they are similar to results obtained with SU(3)-

symmetric versions of the Skyrme model ‘*). In particular we find signs that the

J.A. McGovern, M.C. Birse / Strange baryons (I) 369

cont~butions of strange mesons to baryon properties are over-estimated. This can be corrected by including explicit symmetry-breaking effects, which we will discuss

in a subsequent work i3). We describe the three-flavour hamiltonian and the choice of parameters in sect.

2. Both the SU(3)-symmetric and broken-symmetry versions are described, as well as their hedgehog soliton solutions. In the symmetric case the pions and kaons are degenerate, as are all baryons within a multiplet. The hedgehog solutions of this model are therefore superpositions of members of the octet and decuplet, and the symmetry can be exploited to project out baryon states and calculate their properties. In sect. 3 an approximate projection technique is used to calculate magnetic moments and the ratio g,/gv for certain transitions. The splitting between multiplets can also be calculated using cranking, as described in sect. 4. Some concluding remarks are given in sect. 5.

2. The three-flavour model

The model that we use in this paper to explore properties of the baryon octet is an extension of that which was used for the case of two flavours in ref. 3). There the linear sigma lagrangian of Gell-Mann and Levy 14) was shown to have soliton solutions which could be interpreted as baryons if the fundamental fermion fields represent quarks. In that case the symmetry of the lagrangian is chiral SU(2) x SU(2) and only four meson fields, the pions and the CT, are needed. These transform as a quadruplet under the vector and axial charges. Another independent representation with a scalar triplet and a pseudoscalar singlet could be included, making the symmetry of the lagrangian U(2) x U(2), but as they have no physical counterparts coupling strongly to the nucleon we are free to leave them out. The situation is different in the three-flavour model is). A detailed review of the lagrangian and its properties can be found in ref. 16). Here we summarise the important features to establish notation, and define the procedures we use to determine the model parameters. In this model the mesons belong to the representation (3,3*) + (3*, 3). This contains scalar and pseudoscalar octets and singlets, eighteen mesons in all, and they cannot be split into groups which transform independently under SU(3) x SU(3). We denote the scalar mesons by &, and the pseudoscalar mesons by #_. The physical n and n‘ mesons are not pure octet and singlet, and so will be mixtures of the model &, and &.

The model hamiltonian is

+~(~~~)2+1(71;bn)2+~Tr[VMr.VMJ+U,+c,T,+c,5,},

370 J.A. McGovern, M.C. Birse / Strange batyons (I)

Gell-Mann matrices. For convenience we define the matrix

M = to+ i&+&& + @,)A”.

In terms of M the meson interaction potential is

+ y(det (M) + det ( Mt)) , (2.2)

and the four parameters in this expression together with c0 and c8 are to be fitted

to mesonic masses and decay constants, as described below.

The kinetic and quark-meson interaction terms and U, are SU(3) x SU(3) sym-

metric. The term co&, breaks the symmetry of the hamiltonian (2.1) to SU(3)v and

cs& further breaks it to SU(2)v. The potential has the most general non-derivative

quartic form possible, containing the only three SU(3) invariants which can be

constructed from M. The term in A2 is a generalised Mexican hat potential, and the

second term is required to give large masses to the scalar octet. The term in det (M) models the effects of the instanton-induced U(1) anomaly of QCD “). Without it

the model has an extra U(l)* symmetry, and there is a ninth pseudoscalar Goldstone

boson.

The nine vector currents in this model are

and the axial currents are

(2.3)

(2.4)

where dapy and fapv are the SU(3) structure constants with the addition of da@,,=

&,, and fmpo = 0.

In the case of exact SU(3) symmetry (c8 = 0) the vector currents are conserved,

but for non-zero c8 the currents VE, a = 4,. . . , 7, are not. The model incorporates

PCAC, since the divergences of the axial currents are proportional to the pion (or

kaon) fields. Standard current algebra can be used to obtain the following relations

between the values of the symmetry breaking parameters in the potential and the

vacuum expectation values of the scalar fields to and & (&& and &“), and the

experimental pion and kaon masses and decay constants:

&o,+&8v=-f~, &,+&=m2,f,,

&ov-~&8v=-fK, &,-f&8=m~fK. (2.5)

If c8/c0 were equal to --a, the symmetry would be broken to SU(2) x SU(2) rather

than to SU(2). In fact the values in table 1 give c8/c0= -1.27, indicating that

SU(2) x SU(2) is a better approximate symmetry than SU(3).

The remaining parameters in the potential, A*, K*, v2 and 7, are fixed by the

remaining meson masses, which determine the vacuum expectation values of the

J.A. McGovern, M.C. Birse / Sfrange baryons (I) 371

Parameter fitting for the three-Iiavour modet. The mixing angles 8, and 8. are defined in eq. (2.6)

Summary of parameter

&,, = -127.8 MeV

m, = 533 MeV

h”=41.03

Input masses and decay constants

These determine

A fit to m,, and m,,’ gives

y = 328.6 MeV

which fixes K* = 22.02

and gives

%,=0.123

If 6. is taken as tan-la then

f,=llOMeV

mK = 494 MeV

c,=-16.lOf,m~ &, = 19.63 MeV

m,, = 960 MeV

vz = 6760 MeV’

second derivatives of the potential. In general, &, &, +,, and c,& are not mass

eigenstates for the potential (2.2). These are instead the combinations

71 = cos @,+,,+sin t$&, E = cos @,& + sin e,& ,

77’ = -sin Q$,, + cos @,f$, , 83’ = -sin && + cos 19& , c4w

where 0, and & are the pseudoscalar and scalar mixing angles, respectively.

For a given value of ‘y, and with the pion and kaon masses fixed by experiment,

the values of K* and of the combination h*(&-l- &- v’) are also fixed. Since the

latter is the only form in which A2 and y2 contribute to the pseudoscalar masses,

there is only one free parameter, y, to fit to the rl and q’ masses and the pseudoscalar

mixing angle, 8,. It is satisfactory therefore that good agreement for the masses

and a reasonable mixing angle can be obtained (see table 1).

With y thus determined, either h* and Y* can be taken as a free parameter, and

values of Y’ in the range of -2.0 + 0.4 fmw2 give real scalar masses. Over that range

the lower mass goes from zero to -1200 MeV and the higher from - 1200 MeV to

infinity, while the mixing angle 8, ranges from 79” to 14” (see fig. 1). Such experi-

mental evidence as exists favours the region where the two masses are similar and

there is substantial mixing, and we have worked with values in this range, in particular

the value 0, = tan-’ v’? which gives ideal mixing.

Having fixed all parameters in the lagrangian with the exception of the meson-

quark coupling constant, g, we now want to examine the baryon sector of the model

by looking for soliton solutions. As in ref. 3, we work in the mean-field approxima-

tion 18), in which meson quantum field operators are replaced in the hamiltonian


2000 -

1800-

1600-

1000-

EOO-

Fig. 1. Dependence of the scalar, isoscalar meson masses and mixing angle on the parameter v*. The

other parameters have been fixed as in table 1.

by classical fields, and quantum fluctuations are ignored. In the quark sector only

the valence quarks are included. The energy of the Dirac sea will be altered by the

presence of the soliton, and calculations of this have been done with some effort r9).

It seems unlikely that this will qualitatively change the form of the soliton, at least

for weak enough quark-meson couplings.

Since the strange quarks and mesons have a higher mass than the non-strange

ones, we expect that the lowest-energy solutions will involve only the latter. As in

the SU(2) model, therefore 3), we consider a solution in which the quarks have a

hedgehog form *‘), so only the pion, to and & have non-trivial equations of motion.

The resulting solution will therefore be an eigenstate of grand-spin rather than of

isospin and angular momentum separately. It corresponds to a superposition of the

nucleon and delta, and the parameter g will be chosen to reproduce the average of

their masses. The hedgehog fields are given by

a(r) = a(r) , 4%(r) = thCr) , (2.7)


where i runs from 1 to 3 and &,, and & have been replaced by the ideally-mixed

combinations

~=&J%l+&), s=&&-Jz&3) 7 (2.8)

the first of which couples to non-strange and the second to strange quarks only.

For a static soliton, the mean values of the momenta conjugate to the meson fields

are zero. Including only the above fields, the meson potential U becomes

U =;h2(02+j2+f2- ~~)~+~~~(~~-2~~+~~)~+3~y~(~~+~~)+c,~+c,~, (2.9)

where c, and c, are defined as

C” = &/%, + c*) ) c,=&co-A,). (2.10)

The baryon solution to (2.1) should minimise the expectation of the energy subject

to the normalisation condition j d3rqtq = 1. For a general value of A* the Euler-

Lagrange equations are

VzQi+3qt(igpT~Y,)q-~~=0,

I

v%+3q’gpq-&Y=O,

(2.11)

where the quark wavefunction is a Slater determinant for three quarks described

by a spinor q(r) with the non-strange hedgehog form given in (2.7). The equations

(2.11) can be solved numerically using, for instance, the package COLSYS 2’) which

is well suited to these non-linear coupled differential equations. Convergence can

be checked by comparing the expectation value of the hamiltonian with the Rafelski

virial expression for the energy **). A stable soliton solution is found for all reason-

able values of the scalar masses. If A* is chosen to give ideal mixing, however, the

solution is especially simple, being just an embedding of the solution to the SU(2)

model. To see this, we note that the condition for ideal mixing is that the g (which

will then be the scalar field of the SU(2) model) and the S are mass eigenstates,

that is d* U/~CT&,~ - - 0. From eq. (2.9) this gives the condition

(h2-K2)5”+3&y=0, (2.12)

where lV is the vacuum value of 5. If A2 is chosen to satisfy (2.12), then the

solution to the Euler-Lagrange equation for 5 is simply a constant, 5 = 5”. With the

substitutions


the potential can be written

u = iA &)[ ((+* + +* - v&))* +f(25t - 42,)*1+ dJk+ a” * (2.14)

Up to constant terms this is just the Mexican hat potential of the SU(2) model, and

so the equations for the quark spinor, pion and u fields are exactly the same as in

that model. This embedding of the SU(2) soliton to get a solution to the ideally-mixed

three-flavour model is the same as is used in the SU(3) Skyrme model I’).

Away from ideal mixing, the l field is no longer constant. However, the deviation

from its vacuum value depends only on the radial form of a*++*, which in the

SU(2) case remains near the chiral value of ft for scalar masses above about

700 MeV. Thus it is not surprising that the solitons of the general three-flavour

model do not change significantly in this range either.

The broken symmetry model is the basis for RPA calculations to determine the

mass difference between baryons within the octet, and also for exploration of the

claims made about the strangeness content of the nucleon. It will be discussed in

a subsequent paper 13).

For the remainder of this paper we shall concentrate on the SU(3)-symmetric

version, in which the parameter cs in the hamiltonian (2.1) is set to zero. In addition,

for simplicity the U( l)A breaking term is also discarded by setting y = 0. In this

case the pions and kaons are degenerate, and are given the experimental pion mass

and decay constant; eq. (2.5) then gives co= &tf.. and to” = -&&. Without a

separate kaon mass to fit we have an extra free parameter, and so we can choose

the two scalar masses independently. The mass eigenstates are the to and & fields;

mixing will only occur if they are degenerate. That is if A* = K*, and this is the

condition, a special case of (2.12), under which the hedgehog solution is just an

embedding of the SU(2) one. Again, however, for a Lnge of reasonable scalar

masses the fields of the solution alter very little. We shall use ideal mixing in what

follows, and the form of our solution is therefore the same as in ref. 3), using the

preferred parameter set of that paper: g = 5.38 ( mq = 500 MeV) and m, = 1200 MeV.

3. Projecting the hedgehog

The mean-field solutions to the a-model described so far have all had the hedgehog

form. This has simplified things considerably up to now, but because the solution

breaks the translational, rotational and flavour symmetries of the lagrangian it poses

problems in calculating baryon properties. An infinite class of degenerate solutions

can be obtained by rotating the hedgehog solution in space or isospace. The hedgehog

is analogous to a deformed nucleus, and, as in nuclear physics, Peierls-Yoccoz

projection 23) can be used to obtain states of good spin and isospin. The unprojected

state can be obtained variationally by minimising the excpectation value of the

hamiltonian in either the projected or unprojected state. The former, “variation

after projection”, generally leads to lower energies and hence a better approximation


to the true solution, but the variation must be done separately for each state while

in the latter case it is only performed once.

An exact projection calculation to obtain an estimate of the properties of baryons

in the SU(3)-symmetric a-model would require that the full hedgehog wavefunction

be known. Since we only have the mean fields for the mesons, an approximate

method must be found. Here we have used an extension of the approximate variation

before projection method of Birse and Banerjee ‘) in which the mesonic contributions

to the flavour and angular momentum are ignored. A better approximation would

be to use a coherent state approximation for the meson wave functions, but such a

calculation is involved even in the SU(2) case 24,25). The algebra of SU(3) is

considerably more complex 26) and this approach has not yet been attempted.

Comparison of the SU(2) results shows that the former method tends to overestimate

the quark contributions to the properties by about 15-20%, and underestimate the

meson contributions by about the same amount 24). The final results are rather

similar in both cases.

As already mentioned, the SU(2) hedgehog is not an eigenstate of isospin or

angular momentum. It can however be expressed as a superposition of eigenstates

of these operators, and since the hedgehog has zero grand spin (G = I + J = 0) these

states will have I = J and 1, = -J3. To extend this approach to SU(3) we introduce

flavour eigenstates, each labelled by the representation to which it belongs (for

example 8 or lo), denoted by /.L, and by its hypercharge, isospin and third component

of isospin, collectively denoted by v = ,T,. The hedgehog correlation between the

spin and isospin means that, for the states of interest, we can regard the spin quantum

numbers as describing the isospin of an intrinsic state. The intrinsic quantum numbers

are those of a member of the same SU(3) multiplet p, but with intrinsic or “right”

hypercharge YR= 1. This condition arises because the hedgehog is invariant under

rotations about the hypercharge axis ‘*), so that YR is just the hypercharge of the

unrotated hedgehog, which consists of three non-strange quarks each with hyper-

charge f along with hypercharge-zero mesons. Thus a general state can be written

l/4 v, y’) = IP , I:, > ,‘,J, an d ’ t in erms of these the hedgehog can be expanded as

IHh) = C C(P., 41~ v, -SPY,, (3.1) P”

where -v denotes ,-‘,,, .

The correlation between the spin and flavour means that we need only perform

a flavour projection. Extending the projection to SU(3) we have,

v, v’) = da p(a)D;-&r)d(a)IHh) , (3.2)

where (Y denotes the eight generalised Euler angles, p(a) da is the corresponding

measure for integration 26), and d, is the dimension of the representation p. Here

D~_-y(~) is now an SU(3) D-function, and R(a) is a flavour rotation operator.

376 J.A. McGovern, MC. Birse / Strange baryons (I)

The only operators we shall be using are scalars or third components of vectors.

They belong to a flavour octet, and so are denoted @. For expectation values in

a given state they must be either the third component of an isovector or isoscalar,

that is they are the third or eighth components of the octet with projections p = 1o0

or OoO. For transitions between states in the same representation they will be isospin-

and possibly hypercharge-changing components with p = ,01 or $I;.

All the operators whose expectation values we wish to calculate can be split into

parts which act only on the quarks or only on the mesons. For the quark part,

assuming that radial excitations do not contribute to the nucleon and A matrix

elements, we can write

&““/Hh) = (a&,+ b,@)/Hh) , (3.3)

where ig is a generator of the flavour SU(3) group; b,, = 0 for scalar-isoscalar

operators but in general b, is non-zero otherwise. The coefficients a and b, can be

expressed in terms of the expectation values of 6FjP and ir for the state IHh).

However, the approximation we use consists of ignoring the quantum meson contributions to the wave function IHh) in evaluating these coefficients. The three-quark

piece of the projected wave function must then have the same spin and isospin

structure as the standard quark model, and so the same results will be obtained by

evaluating matrix elements of 0, “q)cL directly between the wave functions of that

model 27).

The evaluation of the matrix elements of meson operators between baryon states

proceeds exactly as in sect. 3 of ref. 3), except that SU(2) ~lebsch-Jordan coefficients for coupling angular momentum are replaced by those governing the coupling of

octets in SU(3). Here also the coefficients C&V) in (3.1) cancel for matrix elements

between the members of the same representation. The final result is

where the reduced matrix element is given by

(3.4)

(3.5)

and &= Sra8r,o. The quantity O”, is the expectation of the meson operator 6”, in

the hedgehog state, and due to the mean-field approximation corresponds only to

the first of the two coefficients in eq. (3.3). The sum over y enters because the octet

representation occurs twice in the product of two SU(3) octets.

It was noted in sect. 2 that the mean-field solution to the SU(3) model is just an embedding of the SU(2) solution given by (2.7). Scalar properties, in particular the

pion-nucleon sigma term, are therefore unchanged. Of the other properties, only

the magnetic moments and the ratio g,/g, for various transitions have been

sufficiently well measured in the octet to provide a test of the model predictions.

J.A. McGovern, M.C. Birse / Strange baryons (I)

magnetic moment a baryon is given

371

/.L~ I

d3r[r i$‘m’]zlB~) , (3.6)

and from (2.3) we have

jg-m’=f~~i(A~+~h*)~-(~,p+~fs~g)(~nai~~+~~~iL). (3.7)

The quark wavefunction is given by eq. (2.7) with xh replaced by xB, the standard

SU(6) spin-flavour spinor, so the quark contribution to the magnetic moment, pu(B9)

is given by

p(gq) =i 2GFr3 dr m’,“’ = 1.743pL,m’,q’, (3.8)

where the nuclear magneton is I.L, = e/2m, and the coefficient rng’ is given in table 2.

For the meson term, the isoscalar and isovector contributions are found by using

(3.4) with 0, A(m)8=jd3r[rA ?b”“]z and p=ooo or loo. The mean-field value of the

operator comes from replacing the meson operators by their mean field values. We

are left with

@““8[Hh) = - $ I

h2(r)r2 dr??( p, ,OO)IHh) = -O’“‘IHh), (3.9)

where 0’“’ = 3.3885~“) and the reduced matrix elements

(8~~~~(m’s~~8~), = -J$jo’m’, (8~~p98~),= -J&O’“‘. (3.10)

These are then multiplied by the appropriate flavour Clebsch-Gordans from (3.4)

to give p(Bm) = O’“‘m(,“) where the rnLm’ are listed in table 2. The presence of charged

strange mesons (kaons) in the projected state results in an isoscalar mesonic contribu-

tion to the magnetic moment. This is in contrast to the SU(2) projection, where the

pions give only isovector terms in the mesonic sector.

The results are given in table 3, both with and without the meson isoscalar

contribution. It can be seen that in most cases the experimental value lies somewhere

between the two, indicating that the SU(3) projection over-estimates this term. This

is not surprising as the expression for the magnetic moment is very sensitive to the

tail of the meson field. Only the kaons contribute to the isoscalar term, and since

in the SU(3) symmetric mode1 they are degenerate with the pions their tail is too long.

TABLE 2

Flavour Clebsch-Gordan coefficients for quark and mesonic matrix elements


TABLE 3

Magnetic moments of the baryon octet

Baryon l-&l1

Magnetic moment (in mm.)

CLmo CLB (/Jr3 - &nO) exp.

P 1.743 0.791 0.113 2.646 (2.534) 2.79

n -1.162 -0.791 0.113 -1.839 (-1.953) -1.91 :+ -0.581 1.743 0.0 0.565 -0.339 0.339 -0.920 2.646 (-0.581) -0.61

(2.308) 2.48 E0 0.581 0.0 0.339 0.920 (0.581) -

z- -0.581 -0.565 0.339 -0.807 (-1.146) -1.1

EJ”o -1.162 -0.226 -0.452 -1.839 (-1.388) -1.25

s- -0.581 0.226 -0.452 -0.807 (-0.355) -0.69

The other quantities calculated are the ratios g,/g, for the weak decays n+ pefi,

A + pefi, X + neP and E- + _4eE In units in which g, for the first transition is one

in the standard quark model, we have for Br + B2

gv= (B,t12 I

d3r VOP(r)lBr?) , &t = (B,‘@ I

d3rMr)hT) . (3.11)

The operators VP and AP which change isospin and flavour by specified amounts

are linear combinations of the vector and axial-vector currents given in eqs. (2.3)

and (2.4). In defining these care must be taken that the choice of phase is consistent

with that used to define the quark wave functions *‘).

Since the meson fields in the soliton are static, only the quarks contribute to gv.

For the quark parts we have

gv = 3 I

(G*+ F*)r* drx&A ‘,ye, = 3,&h Pee, ,

where A ’ is the combination of A matrices which appears in VP.

both quark and meson contributions. The quark part is given by

r

(3.12)

For g,&, there are

&‘=3 J (G*--fF2)r2 drx&u3A pxs, = 3gx&u3A pxB,. (3.13)

The meson operator acting on the internal state gives

&‘)81Hh) = - I( uah+~u~_$?.YT 2

ar r ar

> r dra(p, 1o&-W

= - O’“‘S( p, 1°o)lHh). (3.14)

The reduced matrix elements have the same form as (3.9, and are multiplied by

the appropriate Clebsch-Gordan coefficients from (3.4). The numerical values of g

and 0’“’ are 0.6675 and 2.2428 respectively.


The results are given in table 4. The model predicts axial couplings which are

larger than observed, especially for the nucleon. The use of SU(3) projection does

not provide a significant improvement over the results of the SU(2) model for

this 3,24,25 J.

4. Cranking

4.1. INTRODUCTION

The approximate projection method used in the last chapter cannot be used to

calculate certain quantities of particular interest, namely the mass splittings between

different SU(3) multiplets. Such energy splittings could be obtained from a full

projection calculation 24,25). However, an alternative approach is available which is

familiar from nuclear physics: cranking *).

In this approach one looks for a slowly rotating solution to the time-dependent

equations of motion. The motion is assumed to be purely collective, that is the

internal degrees of freedom are not excited. It can then be described using TDHF *)

(for a deformed nucleus) or the time-dependent mean-field approximation9) (for

a soliton). The cranking equation is obtained by applying the variational principal

to the energy in a frame rotating with angular velocity w, where w acts as a Lagrange

multiplier fixing the mean angular momentum of the state. This cranking equation

can also be derived as an approximation to variation after projection 29), provided

that the system is strongly deformed and the overlaps are sharply peaked functions

of the angles.

Here we work with the lowest-order cranking approximation, assuming that the

rotation is adiabatically slow. We can then treat the cranking term as a first-order

perturbation and the rotational energy is then quadratic in w. The moment of inertia

is defined as twice the coefficient of o2 in the expression for the energy, or

equivalently from the mean angular momentum.

In applying cranking to the quark-meson soliton model the mean-field approxima-

tions used to obtain the static solution are again employed. The mesons are treated

as classical fields, though not this time static ones, so that their conjugate momenta

TABLE 4

Axial couplings for weak transitions between octet baryons

gv ga” gk”’ exp.

n-tp 1 f g &o(m) 1.636 1.25

A-+P -J5 --fig - &dO(m’ 0.967 0.69

Z-+n -Js t&g &GO’“’ -0.372 - 0.36

,.5-+/l 0 2vqg &.BO(m) 0”) 0.01* 0.1 “)

“1 kdg,).


are non-zero. The valence quarks are treated quantum-mechanically, but the Dirac

sea is ignored, as it was in the static calculation. It turns out that due to the symmetries

of the hedgehog 30) the rotating meson fields are unchanged to order o. Hence in

a model such as the Skyrme model where only mesons are present, the moment of

inertia is not altered from its rigid body value, which can be found using the

Adkins-Nappi-Witten collective variable method 3’). In the quark-meson model

the situation is more complex, however, since the quark fields change to first order

in w. These changes must be calculated and their contribution to the moment of

inertia included.

Allowing for the quantisation of spin and isospin gives .% =Jm, and so

the rotational energy becomes $9~~ = $9-‘J(J + 1) = $‘I(1 + 1). Hence the mass

difference between the nucleon (I = i) and the A (I = s) can be determined from the

moment of inertia 3. Cohen and Banerjee 9,30) obtained a value of 1.17 fm for 9,

which leads to a nucleon-A splitting of 252 MeV. This is in good agreement with

the experimental value of 293 MeV. Blaizot and Ripka 32) have shown, in a simplified

chiral model, that lowest-order cranking is probably reasonable for both N and A.

However it is still unclear why cranking gives an N-A splitting which is about twice

that from variation after projection of a coherent-state 24,25).

The isospin moment of inertia is determined by minimising (Q’IH + o * II @‘) with

respect to the small changes in the field caused by the adiabatic rotation. The

rotational symmetry of the hedgehog means that all three isospin axes are equivalent.

Once the SU(2) soliton is embedded in the SU(3) model there is another distinct

moment of inertia: that for strangeness-changing rotations 11*‘2) about axes 4,. . . ,7.

The 8- or hypercharge-axis is an axis of symmetry, and so there is no moment of

inertia associated with rotations about it. If the soliton is cranked about an arbitrary

direction in flavour space with angular velocity w, then we must minimise (0’IH-t

w,Aa I@‘) where A- are the generators of SU(3) expressed in terms of the intrinsic

or “body-fixed” axes. The energy of the cranked state is

E = Eo+t,a,(w:+w:+w:)+q$s(,:+w:+,~+,:), (4.1)

where 9, and LJs are the moments of inertia for isospin (a = 1,2,3) and “strange”

rotations (ff = 4,. . . ,7) respectively. For a particular SU(3) representation, labelled

by (p, q) [ref. 26)] this can be expressed in terms of the Casimir operators of the

angular momentum SU(2) and the flavour SU(3) groups as 1’,‘2)

E((p,q),J, YR)=Eo+@+4i~1)J(J+l)+&‘C2(p,q)-~~a,’Y’,. (4.2)

We have used the fact that (A”) = v$Y~ where Ya is the intrinsic or “right” hyper-

charge discussed in the previous section. Since the hypercharge axis is an axis of

symmetry this is a conserved quantum number which is constrained to be YR = $N,

for N, colours of quark. As noted in sect. 3 the spin J can be regarded as the

intrinsic isospin - the isospin of the members of the multiplet which have Y = +N,.


In these expressions E,, is the energy of the unrotated soliton. This still contains

spurious rotational energy which can only be subtracted in a proper projection

calculation. Hence the cranking energy should only be used to calculate energy

differences between states based on the same intrinsic state.

The mass splittings between different SU(3) multiplets are therefore governed by

the two quantities 9, and 9,. For multiplets like the 8 and 10 which can be constructed

purely from N, quarks, it turns out that only 9, contributes. This can be shown as

follows. For multiplets constructed purely from N, quarks without antiquarks or

mesons one has p + 2q = N, and p = 23 where J is the isospin of the states with

maximum hypercharge (Y = $N,) in the multiplet. For example, for N, = 3, the

permitted values of (p, q) are (3,0) and (2, l), corresponding to the J =s 10 and

J = : 8 respectively; for N, = 5, (p, q) can be (5, O), (3, 1) or (1,2), with J = 5, i, $

respectively. For these states the Casimir operator has the form

C,(p,q)=J(J+l)+&N,(N,+6). (4.3)

Hence the coefficient of 49s in the energy (4.2) is just $N,. This is independent of

J and so the splitting between pure quark multiplets depends only on 4,. The other

moment of inertia, 9s, appears only in the excitation energies of states which must

involve mesonic excitations.

Details of the calculations of 9, and 9s will be given in the following sections.

The results show that 9, is very little changed from its SU(2) value, at 9, = 1.21 fm.

There are certain complications in calculating 9s caused by the non-zero value of

the “intrinsic angular momentum” along the E&axis, but as expected it turns out to

be much smaller, around 0.4 fm. This is consistent with the lack of observed low-lying

states not predicted by the quark model.

4.2. ISOSPIN CRANKING

In order to calculate 9,, we need to minimise the energy of the soliton in a frame

rotating in isospace with angular velocity o, that is to minimise the expectation

value of H’= H +o * I. For sufficiently slow rotations the solution will be only

slightly displaced from the stationary mean-field solution, and we keep only the

changes in the fields which are first order in o. The changes in the meson fields

can be denoted by 4:” etc., and in the quark spinor by q(‘), so that we can write +i = 4:“’ + 4:” and q = q(O) + q(‘)_ Lowest-order cranking will also generate non-zero

conjugate momenta for the meson fields, which we write as GT+,, = 7rgZ’, where nTTm, is

the momentum conjugate to the meson field 4i. Fields not present in the hedgehog

solution, ci, +. and 4*, and their momenta, may also be produced, but rotations

about the isospin axes will not provide source terms for the strange mesons.

The number of fields which have to be considered can be reduced by noting the

symmetry properties ‘) of the cranking term. The isospin operators have grand spin

one, even parity, and are odd under grand reversal (time reversal followed by

382 3.A. McGovern, M.C. Bit-se / Strange baryons (I)

rotation through 7r about the isospin 2-axis). The fields which are non-zero in the hedgehog are all even under grand reversal, and so only those meson fields which vanish in the hedgehog contribute to the changes under cranking. In addition if a field contributes its conjugate momentum does not and vice versa. Thus only the &, &, and #a fields, and the momenta F+~, rrEio and rrx, can be generated by the cranking.

The requirement that the fields have grand spin one and even parity ( GP = 1’) means that & and @s must be in L= 1 states, while & can have L= 0 and 2 components. Similarly the upper components of the spinor qtCf can have L = 0 or 2. The most general form for these fields is

(c)_L A(r)o.a+B(r)(~o.a-~.~Er.ua)

q -4* ( C(r)iw-i-D(r)a-(or\+) > (4.4)

the mean-field we can the expectation of if’ terms of fields and of the soliton, eq. We minimise expectation value + w Z - subject to normalisation constraint the quark This leads a set coupled equations the eight functions introduced (4.4), as as simple for the momenta 9*30).

the aid COLSYS 21) can be straightforwardly. The of inertia found from mean isospin the cranked using

9, -(Z,)/w,. Like therefore it both quark meson contributions:

$‘I’ = &,a ;(q’O’fA,q”‘+ = 3 1 drr*(GA++F(C-20)). wi

The meson contribution is unchanged by the extra fields. The quark part has the same form as in the SU(2) case, but it may be numerically different. In fact it is scarcely changed, as can be seen from table 5. The nucleon-A mass splitting is $%;‘, which has th e value of 245 MeV. Broniowski and Cohen have also carried

TABLE 5

Moments of inertia for isospin cranking, and N-A splitting

9j”’ (fm) J+) (fm) 9, (fm) N-A mass

splitting (MeV)

sue) 0.711 0.461 1.172 252 SU(3) 0.711 0.505 1.216 243


out an SU(2) cranking calculation including vector mesons 33); they found a nucleon-

A splitting of 350 MeV which was hardly changed by the inclusion of n and 5i mesons.

4.3. STRANGE CRANKING, RIGHT HYPERCHARGE AND MONOPOLES

The obvious way to find the other moment of inertia of the SU(3) model, $s, is

to proceed as above for rotations about one of the axes 4,. . . ,7, minimising for

example

(H’)=(H)+w,(A6). (4.6)

However this approach runs into problems, because the hedgehog has a rotational

zero mode corresponding to rotations about the 7-axis and this is not orthogonal

to the source term in the cranking equations “1. Hence the lowest-order cranking

equations do not possess finite solutions. In effect, when one tries to crank about

the 6-axis the system realigns to rotate about the S-axis. A similar problem could

arise in nuclear physics if the intrinsic state had a non-zero expectation value of JZ,

where the z-axis is the symmetry axis. This does not usually occur as states with

J, = +K are degenerate. The cranked state can thus be taken to be an equal mixture

of the two 29) with (JZ) = 0. However, in cases where (JZ) Z 0 the correct procedure

is to crank at an angle to the x- and z-axes 34,35).

Cranking the hedgehog about the 6-axis runs into trouble because the expectation

value of the intrinsic “angular momentum” along the symmetry axis is non-zero:

(A,) = i&Y,. The solution here is to crank at an angle to the 6- and S-axes,

minimising 11)

(H’)=(H-wsin&46-wcospA8), (4.71

where we have written wg = --w sin p and w8 = --w cos p. Adiabatic cranking then

corresponds to the limit of small j?. Since rotation about the 7-axis only alters p

the zero-mode part of the solution can be dealt with separately by minimising (H’)

with respect to p. Consider first a restricted variation, keeping the fields fixed. This

leads to (A,) = 9,,w sin /3 in the cranked state where 40 is the rigid-body moment

of inertia for rotations about the 6-axis. The energy in the rotating frame is

(H’) = &-&P’,,w* sin* /3 -(A& cos p, (4.8)

where (A,) = $fi YR.

Varying (4.8) with respect to /3 to determine the orientation which minimises (H’)

we get

$0w2 sin /3 cos p -(A&J sin p = 0. (4.91

The behaviour of this system has two phases depending on whether o is greater or

384 J.A. McGovern, MC. Birse / Strange baryons (I)

less than the critical value w, = (A,)$,‘. For w < w, the minimum is /l = 0. Since

the rotation axis is aligned with the symmetry axis this does not correspond to a

physical collective motion. When w > w, the situation is different; /3 = 0 is a maximum

and the minimum is at p = cos-r (w,/w). Moreover o, = w cos /3 = W, is constant

and so (A,) = 4,w,; the angular velocity and angular momentum about the symmetry

axis are related by the moment of inertia about the 6- or 7-axis.

The existence of a minimum, quantised angular velocity and momentum about

the symmetry axis, related by the moment of inertia for rotations about a perpen-

dicular axis, is analogous to a charged particle orbiting a magnetic monopole 36).

The collective hamiltonian for the cranked system above, like the monopole system,

has vector-potential terms leading to velocity-dependent forces 35*37). These arise

from the geometric or “Berry” phase 38) which appears when the collective and

intrinsic variables are separated in adiabatic treatments such as cranking. The

monopole-like behaviour induced by the non-zero right hypercharge also occurs in

the Skyrme model 12), where quark quantum numbers are carried by the Wess-

Zumino term. Being linear in time derivatives of the fields, this also leads to

velocity-dependent forces of the type discussed here. The Wess-Zumino term has

a geometric form which can be obtained from an adiabatic approximation to the

fermion action, and it acts like a magnetic monopole in the space of the meson

fields 39).

Variation with respect to the changes in the fields to first order in o sin /3 should

also be done to determine the moment of inertia. However, eq. (4.8) truncated to

second order in sin p will have only the maximum at p = 0. The solution to the

linearised equations of motion will therefore try to align the cranking axis with the

symmetry axis, but such variations have already been accounted for in the variation

with respect to p. Higher-order cranking effects of order p” will alter the value of

p which minimises (4.8) but will not alter the moment of inertia, which simply has

the rigid-body value.

The above approach is a good approximation only when p is small, that is when

the “intrinsic angular momentum” (A”) is large compared with the imposed “collec-

tive angular momentum”, (A”). Th e situation also simplifies when the opposite is

true and /3 = &r. There (A”) is negligible and its effect can be ignored, provided the

part of the solution corresponding to the zero mode is removed.

From the coefficient of 9s in equation (4.2), the collective angular momentum is

(A”)=&,(p, q)-J(J+l)-;Y;. (4.10)

For the purely quark states this goes like a, while (A”) is proportional to N,. In

the limit of large N, one can thus use the first approach and treat the hedgehog as

a rigid rotor.

In work on the SU(3) skyrmion the rigid rotor approximation is used 12). This

cannot be justified by regarding the model as corresponding to the large NC limit

of QCD, since N, = 3 is used in the coefficient of the Wess-Zumino term. However,


we find that with NC = 3 explicit quarks the moment of inertia deviates very little

from its rigid body value, and so this approximation is probably not far off.

4.4. STRANGE CRANKING

To calculate the moment of inertia for “strange” rotations we consider cranking

about the 6-axis. As discussed above the solution will be finite only if the zero-mode

corresponding to rotation about the 7-axis is subtracted from the source.

The fields generated by the cranking in lowest order must have grand spin and

parity GP = i’ and, of course, strangeness *l. However there is no analogue of

grand reversal to guide us in this case. Hence meson fields & and c$~ will be

produced, along with their conjugate momenta. Here the roman indices such as a

run from 4 to 7, while the greek ones run from 0 to 8, as above. Acting on the

hedgehog quark spinor, A6 produces a strange quark with spin up. Hence the

cranked fields will have third component of grand spin MG = +i. Their general

form is

In this case, the cranking equations are obtained by varying (H +d6-- m+bt$)

with respect to the fields and conjugate momenta. The detailed forms for these are

given in the appendix.

Using their solutions and the equations for the momenta, the quark and meson

contributions to the moment of inertia are

92) = -I I

d3r ~(q(“)tA6q’c’+ q(c)th6q(0)) = 3d r2 dr (AG+ BF) . w

(4.12)

The changes in the fields corresponding to an infinitesimal rotation through S

about the 7-axis have the form

(A, 4 64) =$& &, -v’%, h) , (4.13)

and this is a zero mode of the homogeneous equations corresponding to (A.l). In

addition, it is not orthogonal to the inhomogeneous term. Hence the equations as

they stand have no finite solution. In order to extract the finite part of the solution,

the source term for the zero-mode must be removed: the normalised zero-mode


multiplied by its overlap with the source term -g (G, F, 0,O) must be subtracted

from the source. In practice numerical constraints mean that this can never be done

exactly. If a small diagonal regularising term S is added to the equations the response

will always be finite. If the value for the overlap is too small the moment of inertia

will be large and positive; if too large it will be large and negative. By adjusting

the amount of zero mode subtracted one could try to extract the residual value.

Another, more accurate, method is to expand the fields in a Bessel-function basis

with sufficiently large momentum cut-off and density of momenta 4oV4’). The eigenvec-

tors of the homogeneous equations include the zero mode, and one can explicitly

construct the solution to the equations in the space orthogonal to that mode.

018 Wn3-4

O.l6- (0)

0.14-

2.5 3.0 3.5 4.0 4.5 6.0 r lfm)

(b)

2.5 3.0 3.5 4.0 4.5 5.0 r (fm)

Fig. 2. Solution of the strange cranking equations (A.l), (a) q uark radial functions, (b) meson functions.

The radial functions are defined in eq. (4.11). The solution has been orthogonalised with respect to the zero mode (4.13).

J.A. McGovern, M.C. Bit-se / Strange baryons (I) 387

The form of the fields found using this method are shown in fig. 2. The quark

and meson contributions to the moment of inertia obtained $p’= 0.056 fm and

4k”‘,“’ = 0.407 fm giving a total of 4s = 0.463 fm. The lowest-lying excited multiplets

are a J = f lO* at 638 MeV above the nucleon, a J = i 27 at 668 MeV, and a J = i

35 at 860 MeV. Such multiplets, if they exist, contain S = + 1 baryon states and could

show up as resonances in kaon-nucleon scattering. Similar states occur in the

three-flavour Skyrme model and are discussed by Karliner and Mattis 42).

The above procedure is valid if the “intrinsic angular momentum” or right

hypercharge is small compared with the collective or imposed “angular momentum”

about the 6-axis. At the other extreme, where (A”) is large compared with (A”),

cranking should be done at a small angle /3 to the 8-axis. As discussed in sect. 4.3

the quantity to be varied in the rotating frame is then (H - w,A”-wsA8-- E$+$),

where w6 = --w sin p and ws = w cos p = $&YR9S’. To first order in w6 the fields

have the same form as in eq. (4.11). The solution to the equations for them is, as

expected, just the zero-mode associated with rotation about the 7-axis. This corre-

sponds to the realignment of axes which is taken care of by extremising the energy

with respect to p, and so does not contribute to the moment of inertia. This retains

its rigid body value, 9s = 9krn) = 0.407 fm.

For the realistic case N, = 3 the intrinsic angular momentum (A”) is comparable

in size to the collective angular momentum given by eq. (4.11). Hence neither of

the approaches here is entirely correct. However, the difference between the moments

of inertia obtained in these two limits is small, and so even for (A”)-(A”), we

expect 9s to be in the range of 0.41-0.46 fm.

5. Summary

The SU(2) chiral quark-meson model 3,4) has previously been used to calculate

properties of non-strange baryons, and these were found to be in reasonable

agreement with experiment. The model is a linear (+ model 14) describing quarks

coupled to meson fields. It describes the spontaneously broken chiral symmetry

expected from QCD and known to be important for low-energy hadronic physics.

The explicit quark degrees of freedom mean that one is not restricted to local

effective mesonic actions, as in the Skyrme model 6), nor is one tied to large-l\‘,

approximations for calculating baryon properties 31). In the mean-field approxima-

tion the model possesses soliton solutions where three quarks are bound to a

hedgehog configuration of meson fields. This is interpreted as a mixture of physical

baryon states with good spin and isospin. Baryon properties can be obtained from

it with the help of techniques such as projection and cranking 3*24,9).

Since the model embodies the “eightfold way” of current algebra and includes

quarks, it is natural to try to extend it to three flavours. In particular, it is important

to see if it is also capable of a good description of strange baryons. We have extended

the model to include three flavours of quark by using a linear rz model where the


meson fields form a representation of SU(3) x SU(3) [refs. “*i6)]. The representation

which includes pions and kaons is eighteen dimensional, involving nine scalar and

nine pseudoscalar mesons.

If we restrict the interaction potential to be a quartic polynomial in the meson

fields, then the model has seven parameters. The decay constants and masses of the

pions and kaons are experimentally well determined. These fix the magnitudes of

the explicit symmetry-breaking terms in the hamiltonian. In addition they determine

the vacuum expectation values of the scalar fields and so place two constraints on

the remaining parameters. Once these masses and decay constants have been fixed,

the properties of the 7 and 7’ depend only on the coefficient of the term which

breaks the unwanted U(l), symmetry. It is possible to choose this coefficient to

give a good fit the masses of both of these mesons, and their mixing angle.

The properties of the scalar mesons are experimentally rather poorly determined.

However, their masses should be of the order of 1 GeV, and the isoscalar mesons

are expected to be quite strongly mixed. We find a range of values of the remaining

parameter v2 which are consistent with this.

Finally the coupling constant is fixed such that the hedgehog soliton mass agrees

with the average of the nucleon and A masses. This value corresponds to a quark

mass of 500 MeV, which is of the order of magnitude expected for a dynamical

quark mass.

The case of ideal mixing between the scalar mesons is of interest, since a solution

is then just an embedding of the SU(2) hedgehog soliton. Away from ideal mixing,

the embedding is not exact, but provided the scalar mesons have masses above

about 700 MeV the changes in the fields are small.

In this paper we have looked at baryon properties in a simpler version of the

model where the SU(3) symmetry is unbroken. In this limit all the octet baryons

are degenerate, and their properties can be obtained from the hedgehog solution

by means of projection in flavour space. This has been done using an approximate

method ‘) which ignores the contribution of quantum mesons to the matrix elements.

Reasonable results are obtained for the magnetic moments, though the axial coup-

lings for weak decays are about 30% too large, as in the SU(2) model. The deviations

of the magnetic moments show a systematic pattern: the isoscalar mesonic contribu-

tion is generally too large. This is probably due to the tail of the kaon distribution

which, since the kaons are degenerate with the pions, is also too large.

Mass splittings between the various SU(3) multiplets have been calculated using

cranking 8,9), in which we look for solutions to the mean-field equations which are

rotating in flavour space. There are two distinct moments of inertia: one for isospin

rotations and one for “strange” rotations. The former determines the octet-decuplet

splitting. It is extremely close to the value obtained in the SU(2) model 9,30), and

gives a splitting of 245 MeV.

The other moment of inertia appear only in the excitation energies of states which

must involve mesonic excitations. It is much smaller (0.46 fm) and so these states,


if they exist, lie at least 600 MeV above the nucleon. Such states do not occur in

the quark model, and the experimental evidence for them is slight 42).

The calculation of the strange moment of inertia was complicated by the fact that

the hedgehog has a non-zero “intrinsic angular momentum” about the g-axis (or a

right hypercharge of one). Under these circumstances cranking must be done at an

angle to that axis and to the axis about which the moment of inertia is to be

calculated 34,35).

The problem with the isoscalar magnetic moments indicates that the SU(3)-

symmetric model leads to baryons with too large a strange-meson content. Further

indications of this comes from the calculation of quark condensates. If we interpret

the scalar fields &, and & as playing the roles of singlet and octet condensates, then

the projection method of sect. 3 gives essentially the same results as Donoghue and

Nappi found for the SU(3)-symmetric Skyrme model 43). These suggest a surprisingly

large strange quark content in the proton, with consequences which are hard to

reconcile with standard hadronic physics 44). Hence it is important to examine the

effects of SU(3) breaking in this model, in a non-perturbative way. This we do in

a subsequent paper 13), using an RPA approach 8*45).

We are grateful to M.K. Banerjee, T.D. Cohen, F. Lizzi and G. Ripka for useful

discussions. This work is based on part of a Ph.D. thesis submitted to the University

of Manchester and was supported in part by the SERC and a W.W. Smith studentship.

Appendix

The equations for cranking about the 6-axis are obtained by varying (H + wA6 -

&+!I’$) where A6 can be found from (2.3). Variation with respect to the radial

functions introduced in (4.11) gives the equations

A’+(~,+E)B+&~,$F+&~+G= -;$F,

(A.11

where mq = -g&& -v’??&) = gf, for the parameter choice A * = K’ (see sect. 2). The

second derivative of the meson potential a’CJ/a4’ is denoted U,++, and so on:

U,,=h2(5~+5~+h2-~2)+~2(25~+5(5~+h2)-~5850)-6~(5b+~58),

U,, = K*(&&h -&$,h)+3&yh,

U,,=A2(r~+5~+h2-y2)+~2(~(h2+5r~)-~505s)+6~(50+~5s). (A.2)

As mentioned before, we have in this work taken y = 0.


The equations for the meson conjugate momenta again have a simple form,

r+, =_fLiw4i 3 6 =fm& . W.3)

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Strange baryons in a chiral quark-meson model: (I). SU(3)-symmetric model