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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.
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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
systems911 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=ζDζ3and
ξ=r1r2
η=2
3r3r1+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 ηrDq.
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;ζdDq(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;ζ3D(ξ;ζD)δ(Xη)ˆ
Osym(η,η,ξ,ξ;ζ3)
׈
A3nΨDq(η;ζ3D(ξ;ζD)δ(X0η)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=r3r1and ξ00 =r2r3. 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(Xη)ˆ
Osym (η,η,ξ,ξ;ζ3)
׈
A3ΨD(ξ;ζD)δ(X0η)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,X0Dq(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(XX0).(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(XX0)(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
3X1
3X0K(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(XX0)~2
mq2
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
3X02ΨD
2
3X+1
3X0
D
2
3X+1
3X02Ψ
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(XX0),(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|XX0|
3K(E)
ˆ
1(X,X0).(14)
3. Results
3.1. The averaged Nmasses
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,wher=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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If we assume that the strong interactions of baryons and mesons are correctly described in terms of the broken "eightfold way", we are tempted to look for some fundamental explanation of the situation. A highly promised approach is the purely dynamical "bootstrap" model for all the strongly interacting particles within which one may try to derive isotopic spin and strangeness conservation and broken eightfold symmetry from self-consistency alone. Of course, with only strong interactions, the orientation of the asymmetry in the unitary space cannot be specified; one hopes that in some way the selection of specific components of the F-spin by electromagnetism and the weak interactions determines the choice of isotopic spin and hypercharge directions.
Article
A comprehensive treatment of the charmonium model of the ψ family is presented. The model's basic assumption is a flavor-symmetric instantaneous effective interaction between quark color densities. This interaction describes both quark-antiquark binding and pair creation, and thereby provides a unified approach for energies below and above the threshold for charmed-meson production. If coupling to decay channels is ignored, one obtains the "naive" model wherein the dynamics is completely described by a single charmed-quark pair. A detailed description of this "naive" model is presented for the case where the instantaneous potential is a superposition of a linear and Coulombic term. A far more realistic picture is attained by incorporating those terms in the interaction that couple charmed quarks to light quarks. The coupled-channel formalism needed for this purpose is fully described. Formulas are given for the inclusive e+e- cross section and for e+e- annihilation into specific charmed-meson pairs. The influence of closed decay channels on ψ states below charm threshold is investigated, with particular attention to leptonic and radiative widths.
Article
We present a simple parton-model interpretation of the approach to scaling observed in lepton scattering off protons and deuterons. Different final-state configurations are classified and their behavior predicted using quark-counting rules. Good fits to the proton data are obtained. Using a relativistic description of the deuteron, its elastic form factor and inelastic structure function are analyzed. An extraction of the neutron structure function is performed by fitting the deuteron data. Several characteristics of the resulting parametrizations are shown to support our general model. Further experimental consequences are described.