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Baryon-Baryon Interaction in the Strangeness 1 and 2 Sector in the Non-relativistic Quark Cluster Model.

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A N N A L E N D E R PHYSIK
~~
~~~~~
7. Folge. Band 47. 1990. Heft 6, S. 439-518
Baryon-Baryon Interaction in the Strangeness -1
and -2 Sector in the Non-relativistic Quark Cluster Model
By AMANDFAESSLER
and ULRICHSTRAUB
Institut fur theoretische Physik, Universitat Tubingen, FRG
Zurn 200. Jahrestag der Annalen der Physik
A b s t r a c t . The non-relativistic quark cluster model is used to describe the baryon-baryon
interaction in the strangeness -1 sector ( A N , C N system) and in the strangeness -2 sector (Hdibaryon). The calculation in the strangeness -1 sector shows first of all t h a t the cluster model can
be successfully extended from the nucleon-nucleon interaction to the hyperon-nucleon interaction.
I n addition this calculation demonstrates a t the same time the quality of the quark cluster model.
The potentials consist of the one-gluon exchange and the pseudoscalar-meson exchange (both containing central aAd tensor parts), a phenomenological confinement plus :an additional central phenomenological a-meson exchange. Adjusting one free parameter a very good f i t of the experimental
hyperon-nucleon cross sections is achieved. In the second step this model is extended t o the strangeness -2 sector. Here the 13-dibaryon is calculated, which is the most promising candidate for a bound
six-quark state. The H-dibaryon is a linear combination of the three states AA. Z N and XC. Performing the full coupled channel scattering and bound state calculation we predict the H-dibaryon to
be bound. We find a binding energy of (-15 ? 6 ) MeV below the A h threshold, i.e. the mass of the
fi-dibaryon is predicted to be around 2216 MeV.
-
Baryon-Baryon-Wechselwirkung im Strangeness--1- und -2-Sektor
im nichtrelativistischen Quark-Cluster-Mode11
I n h a l t s u b e r s i c h t . Das nichtrelativistische Quark-Cluster-Modell wird zur Beschreibung der
Baryon-Baryon-Wechselwirkung im Strangeness- - I- ( A N , C N System) und --2-Sektor ( H Dibaryon) benutzt. Die Rechnung im Strangeness- - 1-Sektor zeigt zunachst einmal, daB das ClusterModell erfolgreich von der Nukleon-Nukleon-Wechselwirkung auf die Hyperon-Nukleon-Wechselwirkung iibertragen werdcn kann. Daruber hinaus demonstriert diese Berechnung gleichzeitig die
Qualitat des Quark-Cluster-Modells. Die Potentiale bestehen aus dem Ein-Gluon-Austausch und dem
pseudoskalaren Ein-Meson-Austausch (beide enthalten zentrale und Tensorkrafte), einem phlnomenologischen Confinement sowie cinem zusatzlichen phanomenologischen o-Meson-Austausch.
Durch Anpassung des einzigen freien Parameters lrann eine sehr gute Ubereinstimmung mit den
experimentellen Daten der Hyperon-Nukleon-Wechelwirkungerzielt werden. I m zweiten Schritt
wird das Modell auf den Strangeness- -2-Sektor ausgedehnt. Hier wird das H-Dibaryon berechnet,
welches den aussichtsreichsten Kandidaten fur einen gebundenen Sechs- Quark-Zustand reprasentiert. Das H-Dibaryon kann als eine Linearkombination der drei Zustande Ah, E N und CC dargestellt werden. Sowohl eine Streu- als auch eine Bindungsrechnung werden durchgefuhrt. I n beiden
Rechnungen zeigt sich das H-Dibaryon als ein Bindungszustand. Wir finden eine Bindungsenergie
von etwa (-16 & 5) MeV unterhalb der Ah-Schwelle, d.h. die Mmse des H-Dibaryons sollte bei
ungefahr 2 216 MeV liegcn.
Ann. Physik Leipzig 47 (1990)6
440
1. Introduction
Although nuclear physics has been done now for more than fifty years, the elementary baryon-baryon interaction is still an interesting and active field of research. Since
the baryons consist of quarks and gluons, a fundamental understanding of the strong
interaction should be based on these objects. One is convinced that the strong interaction is the residual interaction of the underlying quark-gluon interaction, but a rigorous proof has not yet been given. The reason lies in the running strong coupling constant, which is small only a t high energies, so only there can a perturbative approach
successfully solve the QCD problem. But a t low and medium energies, where one wants
t o understand the strong interaction, this perturbative approach is not justified and
a reliable theoretical treatment of the many-gluon exchange is not available. Therefore
models were developed in order to describe the baryon-baryon interaction a t low and
medium energies.
It seems that among the skyrme model [l],the relativistic quark bag model [a]
and the non-relativistic quark cluster model, the last is the most successful one. It
allows both a quantitative understanding of the properties of single baryons [3] and
of the nucleon-nucleon (NN) interaction [4,5].
It is a very interesting question whether the assumptions of the cluster model made
in order to calculate the NN interaction have general validity. This may be reformulated
by asking whether the cluster model can be extended from the NN interaction t o the
baryon-baryon interaction. To study this problem, we apply the cluster model to the
hyperon-nucleon (YN) interaction and compare our results with the experimental data.
I n the cluster model the dynamics is governed by the Hamiltonian
It is a great merit of the cluster model that the total center-of-mass kinetic energy
KbM can be subtracted exactly. The potentials Vii on the quark level consist of the
one-gluon exchange potential, a phenomenological confinement potential and a pseudoscalar-meson (pion, kaon) exchange potential. The one-gluon exchange and the meson
exchange potentials contain central and tensor parts [6, 71. They are supplemented by
a phenomenological o-meson exchange potential, The strength of the gluanic potentials
is fixed from the mass splitting of the spin-l/2-octet baryons, while the strength of the
pseudoscalar-meson exchange potential is adjusted to reproduce the OBEP value [S].
Only the strength g z / 4 n of the o-meson potential is a fit parameter. The gluonic potentials lead to a short range baryon-baryon force, while the mesonic parts describe the
medium and long range forces.
As orbital wavefunction for a quark i a gaussian ansatz is used :
This ansatz simulates the confinement of the quarks inside the baryons. The size parameter bi is related t o the quark mass according to
_1 b? -
1
1 matrange
mio e3 7
= --.
batrange
b?ight
mlight
(light = U , d )
(3)
This allow a simple separation of the total center-of-mass wavefunction. By antisymmetrizing the six-quark wavefunction the Pauli principle is exactly taken into account.
A. FAESSLER,
U. STRAUB,Baryon-Baryon Interaction
441
The relative wavefunction is calculated with the Resonating Group Method. The details
may be found in ref. 183.
Using this model the hyperon-nucleon interaction (sect. 2 ) and the hyperon-hyperon
interaction (sect. 3 ) is computed. The conclusions are summarized in section 4.
2. Hyperon-Nucleon Interaction
I n this section the main results of the calculation of the AN and C N interaction in
the cluster model are presented.
Contrary to a previous calculation [S], where only s-waves have been taken into
3P1,
account, many more partial waves are now calculated: lS0, IP,, 3S1-3D1,
3P2 - 3P2. Also, besides the central forces [S], the tensor forces due to pion, kaon and
one-gluon exchange are included.
One aim of the previous calculation [S] was to study the effects of varying the parameters of the model. It turned out that the results were not sensitive to the choice of
the size parameter blight. Therefore we will show now also only the results for the size
parameter blight = 0.55 fm. The other important parameter is the strength gz/47z of
o-meson exchange potential. Most experimental data could reasonably well be described
within the range gJ/4n m 3.4-3.8 [8]. Our intention now is to see how closely the present
full calculation can explain the hyperon-nucleon data.
Using Clebsch-Gordan coefficients and taking Coulomb-correction terms, if necessary,
into account we constructed from the computed S-matrices the experimentally measured
cross-sections.
In the Fig. 1 to 5, shown in this section, we compare the results of the previous
calculation [81 where only s-waves are considered with the improved calculation where,
as stated, many more partial waves and the tensor force were taken into account.
The C N system can be coupled to total isospin T = 112 and T = 312, while the
AN system has only total isospin I’ = 1/2. Because of the isospin invariance of the strong
interaction, the CAJ (T = 312) interaction can be computed in a one-channel calculation,
but the A N and ZN (T = 112) interaction has t o be calculated in a coupled two-channel
approach.
2.1. C+p Interaction
The differential and “total” cross-section for the elastic scattering X+p + C+p
is shown in Fig. 1. This is a pure isospin I’ = 312 channel. The tensor forces have only
a very small influence in this channel. The decrease in the differential cross-section a t
backward angles is caused mainly by lS,-lPl and 3S1-3P, interference. From this
calculation the strength g;/43t of the o-exchange is fixed. The value g z / 4 n = 3.34 is
smaller by an almost negligible amount ( 2 % ) than the lower value of ref. [8], which was
g34n = 3.4.
Using this o-meson strength, we calculated the isospin T = 112 AN and CCN channels
without readjusting any parameters.
2.2. A p Interaction
Fig. 2 displays the lambda-proton cross-section a t low energies. Here the C N channel
is closed below 617.5 MeVIc. The lower dashed curve is the cross-section of the previous
calculation, where only s-waves were considered, while the full curve represents the
Ann. Physik Leipzig 47 (1990) 6
442
c+p 3 c+p
150
~
I
.
I
.
I
c’p
~
I
300
.
I
pc+ = l70 MeV/c
bll*,
100
= 3.34
-
full potential
only 5-waves
____._..___
-.......-_ .
250
-
200
-
.
“
“
I
g,2/4rr
-
full potentlol
----._.
1 5 0 4 - 4 + +
100
*
= 3.34
.
only S-waves
...___._.
___----......
50
“
I
3
b
-
0
c’p
blloh,= 0.55 fm
= 0.55 fm
.3,’/4n
-b
---._.
-.-_
I
I
0
--_.
.-...
-
Pig. la,b. Differential and “total” elastic cross-section of I;+p 3 X+p scattering. Data points are
from ref. [lo]
Ap elastic cross section
__________-________
0
100
200
300
400
500
600
700
Fig. 2. Low energy A p -+ A p elastic cross-section. pIabis the momentum of the incident A. Data
points are from refs. [9]
A. FAESSLER,
U. STRAUB,
Baryon-Baryon Interaction
443
result of the present calculation, taking the tensor forces and higher partial waves into
account. The resonance a t the threshold is due to a resonance in the 30, channel, i.e.
the phase shift passes through 90 degrees, but the 3D1S-matrix element itself becomes
quite small just above the threshold.
Another interesting feature of the lambda-nucleon interaction concerns the question
of whether the spin-singlet las or spin-triplet scattering length lat I is larger, i.e.
whether the low energy lambda-nucleon interaction is more attractive in the spinsinglet or spin-triplet state. From the spin assignments of the ground and excited states
of various hypernuclei one is tempted to expect more attraction in the spin-singlet state
than In the spin-triplet state. The cluster model does not support this expectation. We
find more attraction in the spin-triplet state than in the spin-singlet state. This is
caused mainly by the one-gluon exchange and the tensor force. The one-gluon exchange is less repulsive in the spin-triplet state than in the spin-singlet state, while
the meson exchange potentials are almost spin independent. So, even without the
tensor force, one finds more attraction in the spin-triplet state. Inclusion of the tensor force still increases the attraction in the triplet state, but does not influence the
singlet state.
One does not obtain such a clear result in the OBEP models of the lambda-nucleon
interaction. There various version exist which mostly follow the expectation Ja,j >
lat 1, but the opposite relation also sometimes exists [la].
I
2.3. Z p Interaction
I n the Figs. 3-5 the cross-sections for C - p scattering are shown. They are obtained
by combining the isospin channels 7' = 112 and 1' = 312. Here three channels are open
C - p , C - p -+ Con and Z-p 4 An.
and have been measured: 2 - p
Pig. 3 shows the elastic cross-sections C-p --f C - p . The shape of our theoretical
--f
c-p
c-p
-+
c-p
+
c-p
250
b,,,,,
400
200
150
"If
n
n
= 0.55 fm
g,2/4n
= 3.34
300
L
b
100
50
100
0
- 0
1
0
200
0.6
0.2
cos
-0.2
e,
-0.6
-1
120
140
160
b
Fig. 3a,b. Same as Fig. 1, but 2 - p scattering. Data points are from ref.
[lo]
100
444
Ann. Physik Leipzig 47 (1990) 6
results of the differential and total cross-sections seems acceptable but the curves are
generally somewhat too large. The decrease in the differential cross-section a t backward
angles is due t o 3S1-3P0 and 3S1-3P, interference. The influence of the tensor force is
small in this channel.
I n Fig. 4 we present the calculation for the charge exchange reaction X-p + Eon.
Our theoretical curves agree reasonably well with the data. Here no differential crosssection could be measured. The effect of the tensor force again is small.
500-
400
w
300
E
.
I
. , .
= 0.55
b,,,,,
1
.
I
fm
g,2/4n= 3.34
.
-
full potenttal
-...
only S-waver
.._---_--__-._
_..__
_._
Y
200
100
1
0
100
120
.
1
140
.
1
160
.
180
Fig. 4. Total cross-sectionfor the charge exchange reaction Z - p + Eon. Data points are from ref. [ll]
Differential and total cross-sections for the inelastic reaction X-p -+ An are shown
in Fig. 5. It is obvious that in this channel it is impossible t o obtain any reasonable fit
of the experimental data without the tensor force. On the other hand, this channel is
insensitive t o variations of the #-meson strength, since the 0-meson exchange does not
contribute to the transition potential. The largest contribution a t 160 MeVJc to the
.(58%), 3P0 -+ 3P0(17%) and 3S1 3S1(11%).
cross-sections are due t o 3S1+ 301
In summary, in the present version of the cluster model including tensor forces and
higher partial waves, we obtain a very good fit of the various hyperon-nucleon crosssections by adjusting only one parameter, the o-meson strength. The quality of the fit
is now almost as good as in the OHEP calculations [12], which have manymoreadjustable parameters. The improvement of the present calculation as compared with the
previous one is mainly caused by the inclusion of the pion tensor force and the 3S1-3D1
coupling. Nevertheless the main contributions to the cross-sections still come, with
one exception, from the s-waves. So both the central and tensor parts of the forces considered are necessary to understand the hyperon-nucleon interaction. This includes
the one-gluon exchange, the pseudoscalar-meson exchange and the strength of the 0meson exchange.
445
A. FAESSLER,
U. STRAUB,
Baryon-Baryon Interaction
C-p
C p -$ An
+
An
300
b,,,,,
150
g,’/47l
= 0.55 frn
= 3.34
250
200
-3
k
150
b
100
so
0
b
Fig. 5a,b. Differential and total cross-section for the inelastic reaction Z-p + An. Data points are
from ref. [ll]
3. The H-Dibaryon
We have seen that the one-gluon exchange, which is an essential feature of the quark
cluster model, represents an important part of the interaction. For example, the onegluon exchange together with the quark structure of the baryons yields the short range
repulsion both in the NN [4] and the YN [8] interaction.
I n 1977 it was realized by Jaffe [13] that there exist six-quark systems where the
one-gluon exchange yields attraction and not repulsion. Neglecting the mass dependence
of the one-gluon exchange potential, the largest attraction among any six-quark system
due to the one-gluon exchange is obtained if the quarks form a singlet in spin space,
in flavor space and in color space [13,14].
I n flavor-spin space this SU(3)-flavor singlet can be constructed from the baryon
All three two baryon states have the same quark content uuddss. The total spin S and
total isospin T are zero.
It was immediately speculated that this highly symmetric six-quark state might
exhibit a bound state due t o the gluonic attraction. Jaffe, for example, found a binding
energy of 81 MeV using the MIT bag model. He called this state the “H-dibaryon”.
Other calculations using different models find a large variety of binding energies. The
theoretical situation is still very controversial nowadays. Unfortunately, one is presently
not able to decide from experiment whether a bound H-dibaryon exists. A clear experimental result on the H-dibaryon would distinguish which one of these models is “realistic” and would also allow a deeper understanding of the strong interaction. For example, the importance of the one-gluon exchange in the baryon-baryon interaction would
be clarified.
Ann. Physik Leipzig 47 (1990) 6
446
The strong interaction Hamiltonian (1) is isospin invariant but not fully SU(3)flavor symmetric. Consequently the ansatz (4)is only an approximation for the H-dibaryon, i.e. the total spin S = 0 and isospin T = 0 of the H-dibaryon remain good
quantum numbers but the weights of the three channels AR, N E and CC might differ
from the ansatz (4).I n order to investigate the H-dibaryon in the quark cluster model,
we therefore have t o perform a coupled channel scattering and bound state calculation
using this three channels AA, N E and CC [15] with the total spin S = 0 and total
isospin T = 0. The three channels are ordered according to increasing rest mass, i.e.
the A h state has the lowest rest mass of 2 231 MeV. According to the prediction of Jaffe,
the H-dibaryon should be in a IS,,state, for which only the central parks of the potentials
are effective. No tensor forces or spin-orbit forces have to be taken into account.
The results of our scattering calculations can be summarized as follows: The coupling
of the RA t o NS and CC is essential, i.e. if we include only the one-gluon exchange a
resonance is only found in the M JS, phase shift at 20 MeV above the AR threshold if
the coupling of all three channels is taken into account. There is no resonance in the oneand two-channel coupling case, and here the R A phase shift is even repulsive. This
result is almost identical t o ref. [16], but contrary to Jaffe, who found a bound state
including merely the gluonic interaction and ignoring all mesonic contributions. From
the calculations of the NN and YN interaction we know that the inclusion of meson
exchange potentials is absolutely necessary if one wants to obtain a realistic description
of the baryonic forces. Taking these mesonic parts into account a bound state appears
in the scattering calculation, but only in the three coupled channel case.
The properties of the bound state are investigated in a bound-state calculation.
Figure 6 shows the binding energy, taking gluon exchange, pseudoscalar-meson exchange
and the a-meson exchange into account. I present only the case where all three channels
are coupled. I n the one- or two-channel case no bound state is found. Using the same
u-meson strengths as we estimated from the previous calculation [S] of the YN-scattering
the binding energy is roughly 15-20 MeV (Fig. 6) below the RA threshold. If one takes
BINDING ENERGY OF DI-HYPERON VERSUS
SIGMA-MESON COUPLING STRENGTH
20
<
n
10
I
Y
x
P
g
o
-10
9)
g) -20
.T
L
-30
-
- t -
-
2211
0
E
-
Fig. 6. Binding energy of H-dibaryon as function of the a-meson strength. All three channels AA NB - CC are coupled. The range of the o-meson strength obtained from the old fit [S] of the hyperonnucleon data is indicated
A. FAESSLIXL,
U. STRAUB,
Baryon-Baryon Interaction
447
the o-meson strength g2/4n = 3.34 determined from the improved present calculation
including tensor forces, the binding energy should be in the vicinity of 15 MeV below
the threshold. The structure of this bound state [I51 very much resembles the exact
flavor singlet state (4).
Our rcsults are contrary to prcvious calculations (14, 161 done in the cluster model,
which do not find a bound H-dibaryon. The rcason lies mainly in the mesonic potentials,
which are neccssaiy to describe the N N and YN data but which were not included in
rc.fs. [ 14, I GJ. Therefore the calculations of refs. r14, 161 cannot be viewed as “realistic”.
Wtl predict the binding energy of the H-dibaryon with respect to the A h threshold of
2M,\ = 2231 MeV to be (-15 & 5) MeV. Thus the mass of the H-dibaryon is (2216
5) McV. The cstimatcd uncertainty has its origin partly in the choice of the fundamental
size parameter bllnh+[ 151 but mainly in the strength of the o-meson potential.
+
4. Conclusions
The non-relativistic quark cluster modcl has been successful in describing the nucIcon-nuclcor~interaction. In the present paper we have presented the results of extending
thcsc calculations to the hyperon-nucleon interaction and the H-dibaryon. The quark
vliistcr modcl incorporated short range forccs (gluonic potentials), medium range forces
(kaori, o-meson exchange) and long range forces (pion exchange). The central arid the
tciisor parts of the various potentials were taken into account. Fitting one parameter,
thc strength of the o-meson exchange, a series of hyperon-nucleon cross-sections was
calcrilntcd. The agreemcnt between our theoretical results and the experimental data
was surprisingly good. The quality of the fit was almost as nice as in the OBEP calculations which had many more adjustable parameters. The improvement of the present
calculation including tensor forces as compared with the previous one without tensor
forcvs was mainly caused by the inclusion of the pion tensor force and the 38x1-3D,
c~orrpliug.I b t h were necessary to understand cspecially the data in the channel X-p 3
A n . hi the other hyperon-nucleon channels the s-wave cross-sections represented still
the nia jor contribution to the cross-sections. Applying this model to the H-dibaryon,
a coupled system of the threc channels A A , EN and ZX, the appearance of a bound state
was predicted. The binding energy of the H-dibaryon with respect t o the AA threshold
of 2 M A = 2 2 3 I MeV was estimated to be (- 15 & 5) MeV. Thus the mass of the H dibaryon was (2216 & 5) MeV. The uncertainty had its origin mainly in the strength
of the c-mcson potential.
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Ann. Physik Leipzig 47 (1990) 6
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Boi der Redaktion eingegangen am 9. November 1989.
Anschr. d. Verf.: Prof. A. FAESSLER
Dr. U. STRAUB
lnstitut fur theoretische Physik
Universitit Tiibingen
D-7400 Tiibingen
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