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February 26, 2002 12:25 WSPC/143-IJMPE 00024

International Journal of Modern Physics E, Vol. 11, No. 1 (2002) 35–44

c

World Scientiﬁc Publishing Company

THE EFFECT OF ANTISYMMETRIZATION IN

DIQUARK MODELS OF BARYONS

B. R. MABUZA

Department of Physics, University of South Africa, P.O. Box 392,

Pretoria 0001, South Africa

R. M. ADAM∗

Department of Arts, Culture, Science and Technology,

Private Bag X894, Pretoria 0001, South Africa

B. L. G. BAKKER

Fakulteit Natuurkunde en Sterrenkunde, Vrije Universitieit,

De Boelelaan 1081, NL-1081 HV Amsterdam, The Netherlands

Received 1 November 1998

The eﬀect of antisymmetrization in diquark models of baryons composed of light (u&d)

quarks is investigated. The diquark in this study is considered alternately as a point-like

and as a composite particle where antisymmetrization is taken into account by means

of Generator Coordinate Model operator kernels. The eﬀect on ground state masses and

form factors is striking and we are able to conclude that there is a strong dynamical

eﬀect due to the presence of antisymmetrization in diquark models of baryons.

1. Introduction

The diquark system is formed as a bound state of two quarks and regarded as

elementary when it interacts with the third quark (spectator) to form a baryon.

The diquark simpliﬁcation is as old as the quark hypothesis itself and was suggested

by Gell-Mann in his pioneering paper on quarks.1Since Gell-Mann’s suggestion that

a bound state of two quarks may be stable, a number of authors2have considered

multiquark models containing diquarks. Close3proposed that the attraction which

a spectator quark feels to the remaining pair of valence quarks in a baryon involves

the same eﬀective colour charge as between this quark and an antiquark.

In particular, a model of excited baryon states in which one quark acquires or-

bital angular momentum while the other two quarks remain as an S-wave diquark

has been useful.4Leinweber5and Fleck et al.6have performed theoretical investi-

gations of diquark clustering which produced encouraging results. However, most

∗Please address all correspondence to this author at adam@acts2.pwv.gov.za.

35

February 26, 2002 12:25 WSPC/143-IJMPE 00024

36 B. R. Mabuza, R. M. Adam & B. L. G. Bakker

authors have simply assumed that diquarks can be treated as elementary, without

any attempt being made to antisymmetrize the baryon wave function.7Antisym-

metrization is necessary because of the exchange of cluster quarks with the spectator

inside baryons. Nevertheless the complications involved in antisymmetrization re-

sult in the loss of some of the values of the simple diquark picture. Lichtenberg7

claims that the good predictions obtained in calculations using diquarks without

antisymmetrization demonstrates that the interference terms resulting from anti-

symmetrization must be relatively unimportant. Nevertheless, the only way to show

that these terms are indeed small is to antisymmetrize the wave function.

The motivation for undertaking the calculations presented here was to examine

the eﬀect of antisymmetrization in diquark models of baryons in the ground state

within a well-deﬁned diquark model. The Generator Coordinate Model (GCM)8

operator kernels provide a transparent and explicit way for including antisym-

metrization within such a model. The GCM has been applied primarily in nuclear

systems9–11 but is perfectly general, with the generator coordinates standardized as

the intercluster vectors. In our work the expectation values for several observables

have been calculated in a pure diquark model and compared with corresponding re-

sults where antisymmetrization has been included via the GCM kernels. The results

are also compared with those for three-body models of baryons and with experi-

ment. The observables studied were the masses, the elastic electric form factors and

the charge radii. The diquark model used was a simple non-relativistic one involving

central interactions and a single channel.

The antisymmetrization formalism is laid out in Sec. 2, the results are presented

and discussed in Sec. 3 and the conclusion and prospects are discussed in Sec.4.

2. Formalism

2.1. The anti-symmetrization scheme

We begin by assuming 1) that the dynamics of the three quark system are well-

approximated by a two-body Dq model (where Ddenotes the diquark and qthe

spectator quark) and, 2) that we have as primary input for our antisymmetrization

procedure, a converged wave function ΨDq (rDq;ζB), where rDqis the vector be-

tween the centre of mass of the diquark and the spectator quark and ζBrepresents

a set of spin, isospin and colour quantum numbers associated with the baryon. This

converged wave function is obtained in the following manner:

1. The ground state wave function (and hence the mass) of the diquark is obtained

by solving the Schr¨odinger equation for a particular quark-quark interaction.

2. The mass obtained in (1) is used as an input into the Schr¨odinger equation for

the diquark-quark system using the same quark-quark interaction.

In this work we use central quark-quark potentials, which implies that our di-

quarks are a superposition of vector and scalar diquarks and that our baryons are

spin/isospin-weighted averages over the appropriate multiplet — e.g. N−∆ average

February 26, 2002 12:25 WSPC/143-IJMPE 00024

The Eﬀect of Antisymmetrization in Diquark Models of Baryons 37

rather than p,n,∆

+etc. This simpliﬁcation reduces the system of equations from

a multi-channel to a single channel framework.

To treat the diquark as a composite particle rather than as a point particle we

replace the two-body wave function ΨDq(rDq;ζB) with the wave function

ΨB(ξ,η;ζB)=Ψ

D(ξ;ζD)⊗ΨDq(η;ζB),(1)

where ζB=ζD⊗ζ3and

ξ=r1−r2

η=2

√3r3−r1+r2

2(2)

are Jacobi coordinates (with particles 1 and 2 forming the diquark and particle 3 the

spectator quark) and ΨD(ξ;ζD) is the diquark wave function obtained according

to the procedure described above. Note that η∝rDq.

Although ΨDand ΨDq have the correct symmetry individually, ΨBis not fully

anti-symmetrized at the three-body level. To anti-symmetrize ΨBwe divide the

system into two clusters, namely the diquark Dand the spectator quark q.

In the two-body (diquark) model of a baryon Bthe expectation value of an

operator ˆ

O(η,∇η;ζd)isgivenby

hˆ

Oi=1

hˆ

1iZΨ†

Dq(η;ζ3)ˆ

O(η,∇η;ζd)ΨDq(η;ζ3)d3η, (3)

where the integration involves spin and isospin averages as well and where hˆ

1iis

the normalization.

In the three-particle scheme Eq. (3) is replaced with

hˆ

Oi=1

hˆ

1iZΨ†

Dq(η;ζ3)ΨD(ξ;ζD)δ(X−η)ˆ

Osym(η,∇η,ξ,∇ξ;ζ3)

×ˆ

A3nΨDq(η;ζ3)ΨD(ξ;ζD)δ(X0−η)od3ηd3ξd3Xd3X0,(4)

where ˆ

A3is the antisymmetrizer for the three quarks and where ˆ

Osym is a gener-

alized and symmetric version of the operator ˆ

O. For example, the quark-diquark

potential operator 2V(η) would be replaced by the operator V(ξ)+V(ξ0)+V(ξ00),

where ξ0=r3−r1and ξ00 =r2−r3. The operator ˆ

A3results in one direct term and

two exchange terms, corresponding to the possible permutations of quarks through

the diquark and spectator clusters.

We now use Eq. (4) to deﬁne the operator kernel Kˆ

Ocorresponding to the

operator ˆ

O:

Kˆ

O(X,∇X,X0,∇X0)=ZΨD(ξ;ζD)δ(X−η)ˆ

Osym (η,∇η,ξ,∇ξ;ζ3)

×ˆ

A3ΨD(ξ;ζD)δ(X0−η)d3ηd3ξ. (5)

February 26, 2002 12:25 WSPC/143-IJMPE 00024

38 B. R. Mabuza, R. M. Adam & B. L. G. Bakker

Expressing the expectation value hˆ

Oiin terms of Eq. (5) we ﬁnd:

hˆ

Oi=1

hˆ

1iZΨ†

Dq(X;ζ3)Kˆ

O(X,∇X,X0,∇X0)ΨDq(X0;ζ3)d3Xd3X0.(6)

Application of Eq. (5) to the norm, kinetic energy, potential energy and

form factor operators yields the following expressions for the operator kernels for

spin/isospin averaged S-states within the N−∆ multiplet:

Norm

Direct term:

K(D)

ˆ

1(X,X0)=δ(X−X0).(7)

Exchange term:

K(E)

ˆ

1(X,X0)=22

√33

Ψ†

D

1

√3X+2

√3X0ΨD

2

√3X+1

√3X0,(8)

where ΨD(r) is the spatial component of the diquark wave function.

Potential energy — pure central force

Direct term:

K(D)

ˆ

V(X,X0)=δ(X−X0)(hViD+2Zd3ξ|ΨD(ξ)|2V

1

2ξ+√3

2X!),(9)

where hViDis the expectation value of the potential energy operator within the

diquark subsystem.

Exchange term:

K(E)

ˆ

V(X,X0)=V

1

√3X+2

√3X0+V

2

√3X+1

√3X0

+V

1

√3X−1

√3X0K(E)

ˆ

1(X,X0),(10)

where K(E)

ˆ

1(X,X0) is the exchange part of the norm kernel.

Kinetic energy

Direct term:

K(D)

ˆ

T(X,X0,∇X)=δ(X−X0)−~2

mq∇2

X+hˆ

TiD,(11)

where hˆ

TiDis the internal kinetic energy of the diquark subsystem.

February 26, 2002 12:25 WSPC/143-IJMPE 00024

The Eﬀect of Antisymmetrization in Diquark Models of Baryons 39

Exchange term:

K(E)

ˆ

T(X,X0,∇X,∇X0)

=−~2

2mK(E)

ˆ

1(X,X0)∇2

X+∇2

X0

+2

√33Ψ†

D

1

√3X+2

√3X0∇2ΨD

2

√3X+1

√3X0

+Ψ

D

2

√3X+1

√3X0∇2Ψ†

D

1

√3X+2

√3X0 .(12)

Form factor

For S-states, and neglecting the meson cloud at this point, the kernel corresponding

to the elastic electric form factor operator G(Q2) reduces to:

Direct term:

K(D)

ˆ

G(X,X0,∇X)=1

3(j0QX

√3

+2Zd3ξ|ΨD(ξ)|2j0 Q

√3

2ξ+1

2X!)δ(X−X0),(13)

where j0(x) is the spherical Bessel function of order zero.

Exchange term:

K(E)

ˆ

G(X,X0,∇X,∇X0)= 1

3j0QX

√3+j0QX0

√3

+j0Q|X−X0|

√3K(E)

ˆ

1(X,X0).(14)

3. Results

3.1. The averaged N∆masses

In the case of central potentials, the implicit averaging over spins necessitates a

comparison with the averaged measurements rather than with those of a speciﬁc

baryon. In general, two-body models of baryons produce unphysically deep binding

due to the large reduced mass of the quark-diquark system. The resulting masses

are about 180 MeV lower than the corresponding three-body results for the same

potentials — see Table 1.aThis trend can be understood in terms of the fact that

aNote: The constituent quark masses used are deﬁned in the potential models and are 337 MeV,

330 MeV and 300 MeV for the Bhaduri,14 Cornell13 and Martin15 potentials respectively.

February 26, 2002 12:25 WSPC/143-IJMPE 00024

40 B. R. Mabuza, R. M. Adam & B. L. G. Bakker

Table 1. Averaged N−∆ masses calculated using (i) the diquark

model without antisymmetrization; (ii) the diquark model includ-

ing Resonating Group Model (RGM) antisymmetrization; (iii) a full

three-body treatment using the IDEA method12 for the Cornell,13

Bhaduri14 and Martin15 interactions. All masses are in MeV.

Interaction Diquark Diquark with RGM Three-body via IDEA

Cornell 913.5 1117.8 1089.5

Bhaduri 1024.1 1235.0 1204.0

Martin 907.1 1122.9 1086.0

Table 2. Potential and kinetic energy contributions to the averaged N−∆

mass calculated using the Cornell,13 Bhaduri14 and Martin15 within the di-

quark model without antisymmetrization. All energies and masses are in MeV.

Interaction Potential Energy Kinetic Energy Binding Energy Mass

Cornell −606.9 400.7 −206.2 913.4

Bhaduri −550.9 402.3 −148.6 1024.1

Martin −525.6 390.6 −135.0 907.1

Table 3. Potential and kinetic energy contributions to the averaged N−∆

mass calculated using the Cornell,13 Bhaduri14 and Martin15 within the di-

quark model augmented by RGM antisymetrization. All energies and masses

are in MeV.

Interaction Potential Energy Kinetic Energy Binding Energy Mass

Cornell −647.5 775.3 127.8 1117.8

Bhaduri −559.2 783.2 224.0 1235.0

Martin −480.7 703.6 223.1 1123.1

both the inclusion of the binding energy in the diquark mass and the “reduction”

in the number of particles (from three to two) in the system are non-variational

procedures. For the central potentials used the two-body masses range between

907.1 MeV and 1024.1 MeV (the experimental value is 1173 MeV). The corre-

sponding three-body results vary from 1086 MeV to 1204 MeV and were calculated

by means of the Integrodiﬀerential Equation Approach (IDEA).12 For three bod-

ies the IDEA is an augmented version of the S-projected Faddeev Equation and

takes higher partial waves into account in an average way. For interactions with

no hard core, such as those encountered in quark systems, the IDEA is essentially

exact.

There is much better agreement between the three-body IDEA masses and the

equivalent two-body GCM kernel results than when antisymmetrization is not taken

into account. The agreement for all three potentials considered is within about four

percent. In all cases, the three-body masses are somewhat lower — see Table 1.

February 26, 2002 12:25 WSPC/143-IJMPE 00024

The Eﬀect of Antisymmetrization in Diquark Models of Baryons 41

Table 4. Diquark energy expectation values and masses calculated using the

Cornell,13 Bhaduri14 and Martin15 interactions. All masses are in MeV.

Interaction Potential Energy Kinetic Energy Binding Energy Mass

Cornell −124.0 253.6 129.6 835.7

Bhaduri −93.1 254.7 161.7 789.6

Martin −63.2 205.3 142.1 742.1

Table 5. Root-mean-square charge radii using the Cornell,13 Bhaduri14

and Martin15 interactions within the pure diquark model without antisym-

metrization, and the RGM-augmented diquark model (each with and without

a meson cloud contribution). Radii are in fm. The experimental value for the

proton is 0.83 fm. Diquark = dq, meson cloud = mc

Interaction Pure dq Pure dq + mc RGM dq RGM dq + mc

Cornell 0.40 0.73 0.46 0.76

Bhaduri 0.39 0.72 0.45 0.76

Martin 0.43 0.74 0.51 0.79

Given the fact that the kinetic and potential energies are appreciable fractions of

the masses this concordance is noteworthy.

Comparing results in Tables 2 and 3 we see that the antisymmetrization proce-

dure has relatively little inﬂuence on the expectation values of the potential energy.

The large positive shifts in the masses are entirely due to the approximate doubling

of kinetic energy expectation values. Are these shifts believable? Gavin et al.,16

using the hypervirial approach17 in a three-body system, have determined that the

expectation value of the kinetic energy is about 70% of the baryon ground state

mass for quark potential models. This result corresponds very well with the GCM

diquark values in Table 4.

The relative model independence of the potential energy (two-body results

including and excluding antisymmetrization are not very diﬀerent from the

corresponding three-body values) is interesting. The explanation for this insensi-

tivity is probably that the motion of the third quark is mediated by an interaction

which is well-approximated by an aggregate static potential resulting from the other

two quarks. This explanation is closely related to the reason for the success of the

non-relativistic quark potential model: the colour ﬁelds are essentially frozen with

respect to the motion of the quarks.

The explanation for the low kinetic energy expectation values when antisym-

metrization is not taken into account via the GCM kernels must be that the large

values of the quark-diquark reduced masses (see Table 5) produce unphysically deep

binding for the eﬀective two-body system. This eﬀect is oﬀset when the GCM ki-

netic energy operator, which is a function of constituent quark masses rather than

diquark masses, is used.

February 26, 2002 12:25 WSPC/143-IJMPE 00024

42 B. R. Mabuza, R. M. Adam & B. L. G. Bakker

3.2. Form factors

The elastic electric form factor results (see Fig. 1) conﬁrm the pattern of better re-

production of experimental data when we introduce the antisymetrization kernels.

When the eﬀect of the meson cloud around the constituent quarks is introduced by

means of the appropriate monopole factor Λ2

Λ2+Q2,whereΛ=0.795 GeV, the cor-

respondence between the proton data (taken from Hohler20)andtheGCMkernel

results is almost perfect. This type of interpolation between the low Q2meson-

dominated limit and the high Q2quark-dominated limit has been motivated by

various authors.18 To be consistent with our overall spin-independent approxima-

tion, we have excluded additional contributions such as that of extended vector

meson dominance. This approach has yielded good results in three-body calcula-

tion of baryon form factors.16

q fm 1

FEq

14121086420

1

0.8

0.6

0.4

0.2

0

Fig. 1. Electric form factor of the proton. Dotted line = pure diquark, dashed line = anti-

symmetrization included, full-line = antisymmetrization and meson cloud eﬀects inluded, + =

experimental data. The calculations were done using the Cornell potential.

February 26, 2002 12:25 WSPC/143-IJMPE 00024

The Eﬀect of Antisymmetrization in Diquark Models of Baryons 43

The agreement between the results of calculations of r.m.s. charge radii is also

improved by the introduction of antisymmetrization. When the meson cloud con-

tribution is included (see above), the r.m.s. radii agree to within 2 — 5% of the

experimental proton value of 0.83 fm. This is further evidence of the usefulness of

the diquark model improved by antisymmetrization. The relative sizes of (i) pure

diquark (∼30%), (ii) antisymmetrization (∼10%) and (iii) meson cloud (∼60%)

contributions to the square of the charge radius also give an indication of the mag-

nitude of the eﬀect of antisymmetrization.

4. Conclusion

It is clear from this study that diquark wave functions may be used to extract phys-

ical information which is of a surprisingly high quality. This aﬃrms the wide appli-

cability of the GCM and more generally the resonating group model approaches.

These were developed more speciﬁcally for few-nucleon systems (see, for example,

Refs. 9 and 11) but we have shown here that the GCM is equally applicable to

quark systems.

A logical generalization of this work would be to extend the analysis to systems

of coupled channels. The single channel model implies a comparison with a ﬁctitious

system rather than a proton or a neutron. Analysis of other baryons and of excited

states would also be possible then.

Another interesting way to build on the formalism developed here would be to

calculate relativistic corrections to the masses. This could be done by transforming

the diquark and quark-diquark wave functions to momentum space and calculating

the corresponding kinetic energy expectation values using the appropriate relativis-

tic operators. To include relativistic eﬀects in coordinate space is not convenient

because of the square root form of the kinetic energy operator. However, the ap-

proach described above would also not give the full relativistic correction, which

also arises from a consideration of retardation eﬀects and of Lorentz invariance

of the appropriate phase space. These eﬀects are most easily treated in momen-

tum space too. Nevertheless, conﬁning potentials (which necessarily occur in quark

calculations) provide some diﬃculty in momentum space. Recently, however, some

progress has been made in applying the fact that the Fourier transform of a linearly

increasing potential does exist in a distributional sense21 and it would be interest-

ing to apply this full formalism (hitherto restricted to mesons) to diquark models

of baryons too.

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