Quark-model based study of the triton binding energy
B. Julia
´
-Dı
´
az,
1
J. Haidenbauer,
2
A. Valcarce,
1,3
and F. Ferna
´
ndez
1
1
Grupo de Fı
´
sica Nuclear, Universidad de Salamanca, E-37008 Salamanca, Spain
2
Institut fu
¨
r Kernphysik (Theorie), Forschungszentrum Ju
¨
lich, D-52425 Ju
¨
lich, Germany
3
Departamento de Fı
´
sica Teo
´
rica, Universidad de Valencia, E-46100 Valencia, Spain
共Received 18 September 2001; published 6 February 2002兲
The three-nucleon bound state problem is studied employing a nucleon-nucleon potential obtained from a
basic quark-quark interaction in a five-channel Faddeev calculation. The obtained triton binding energy is
comparable to those predicted by conventional models of the NN force.
DOI: 10.1103/PhysRevC.65.034001 PACS number共s兲: 12.39.Jh, 21.45.⫹v, 13.75.Cs, 21.10.Dr
I. INTRODUCTION
During the last decade the development of quark-model
based interactions for the hadronic force has led to nucleon-
nucleon (NN) potentials that provide a fairly reliable de-
scription of the on-shell data. Several models including
quark degrees of freedom have been used to study the NN
interaction 关1兴 and also the baryon spectra 关2兴. Among them,
the chiral quark cluster model is the only one that pursued a
simultaneous understanding of different low-energy phenom-
ena based on a unique quark-quark interaction. This model is
able to provide a quantitative description of the NN scatter-
ing 关3兴 and bound state problems 关4兴.
Nevertheless, quark-model based NN interactions have
not been often used to study few-body systems. There might
be two different reasons for that. First of all, most of those
interaction models for the two-nucleon system needed to be
supplemented with meson-exchange potentials between the
baryons to obtain a reasonable description of the experimen-
tal data 关5,6兴, losing in this way their quark based character.
Second, other quark-model based interactions were primarily
designed to describe the baryon spectra 关7兴, but lead to un-
realistic results when they are applied to the two-nucleon
system 关8兴.
In this work we want to perform a study of the triton
bound state making use of a nonlocal NN potential fully
derived from quark-quark interactions. The model has been
previously utilized for investigations of three-body systems
(NNN, NN⌬, N⌬⌬, and ⌬⌬⌬), putting more emphasis on
the mass ordering of possible bound states of these systems
than on the binding energy values 关9兴. In the present work
the full nonlocal NN potential will be employed as it follows
from the application of the resonating group method 共RGM兲
formalism. This method allows, once the Hilbert space for
the six-body problem has been fixed, to treat the intercluster
dynamics in an exact way. Thereby, nonlocalities are gener-
ated, reflecting the internal structure of the nucleon, which
translate into specific off-shell properties of the resulting NN
potential.
Indeed the relevance and/or necessity of considering the
nonlocal parts of NN potentials in realistic interactions is
still under debate. Over the past few years several studies
have appeared in the literature which stress the potential im-
portance of nonlocal effects for the quantitative understand-
ing of few-body observables and, specifically, for the triton
binding energy 关10–15兴. However, the majority of these in-
vestigations 关11–15兴 explore only nonlocalities arising from
the meson-exchange picture of the NN interaction.
The nonlocalities generated in a quark-model derivation
of baryonic potentials may play a relevant role for the case of
the three-nucleon bound state. It has been argued that the
assumptions associated with meson-exchange models
sharply limit the nature of the off-shell properties of those
potentials, once the on-shell matrix elements are constrained
to fit the NN data 关16兴. Therefore, it is very interesting to
investigate the off-shell features of potentials derived from a
quark model. Some preliminary studies in this direction have
been done in Ref. 关10兴. However, there only the short-range
part of the interaction is obtained by means of quark-model
techniques. The intermediate- and long-range parts are de-
scribed by ‘‘standard’’ meson exchange between baryons.
Accordingly, that work allows only very limited conclusions
with regard to effects of the quark substructure.
The triton binding energy is obtained from a Faddeev cal-
culation. We restrict ourselves to the standard five-channel
case; i.e., we consider only the
1
S
0
and
3
S
1
-
3
D
1
NN partial
waves, those which provide the bulk contribution to the
three-nucleon binding energy. The three-body Faddeev equa-
tions will be solved in momentum space, making use of
separable finite-rank expansions of the two-body interac-
tions.
The paper is organized as follows. In Sec. II we introduce
the basic quark-quark Hamiltonian and we describe the
method to obtain the RGM NN interaction. In Sec. III we
provide details about the finite-rank expansions of the quark-
model based potentials which enter in the Faddeev calcula-
tions of the triton binding energy and we present results for
the three-body system. Finally, some concluding remarks are
provided in Sec. IV.
II. QUARK-MODEL BASED NN POTENTIAL
In recent years a chiral quark cluster model for the NN
interaction has been developed. This model has been widely
described in the literature 关3,4,17,18兴; therefore, we will only
briefly summarize here its most relevant aspects. It contains,
as a consequence of chiral symmetry breaking, a pseudo-
PHYSICAL REVIEW C, VOLUME 65, 034001
0556-2813/2002/65共3兲/034001共5兲/$20.00 ©2002 The American Physical Society65 034001-1
scalar and a scalar exchange between constituent quarks
coming from the Lagrangian
L
ch
⫽ g
ch
F
共
q
2
兲
⌿
¯
共
⫹ i
␥
5
ជ
•
ជ
兲
⌿, 共1兲
where F(q
2
) is a monopole form factor:
F
共
q
2
兲
⫽
冋
⌳
2
⌳
2
⫹ q
2
册
1/2
. 共2兲
⌳
determines the scale of chiral symmetry breaking, being
bound between 1 GeV and 600 MeV 关19兴. The chiral cou-
pling constant g
ch
is chosen to reproduce the experimental
NN coupling constant.
From the above Lagrangian a pseudoscalar (PS) and a
scalar 共S兲 potential between quarks can be easily derived in
the nonrelativistic approximation:
V
ij
PS
共
q
ជ
兲
⫽⫺
g
ch
2
4m
q
2
⌳
2
⌳
2
⫹ q
2
共
ជ
i
• q
ជ
兲
共
ជ
j
• q
ជ
兲
m
PS
2
⫹ q
2
共
ជ
i
•
ជ
j
兲
, 共3兲
V
ij
S
共
q
ជ
兲
⫽⫺g
ch
2
⌳
2
⌳
2
⫹ q
2
1
m
S
2
⫹ q
2
. 共4兲
Using the range of values for ⌳
given above yields a
N⫺ ⌬ mass difference due to the pseudoscalar interaction
between 150 and 200 MeV. The rest of the mass difference,
up to the experimental value, must have its origin in pertur-
bative processes. In the present model, this is taken into ac-
count through the one-gluon-exchange potential 关20兴
V
ij
OGE
共
q
ជ
兲
⫽
␣
s
共
ជ
i
•
ជ
j
兲
再
q
2
⫺
4m
q
2
冋
1⫹
2
3
共
ជ
i
•
ជ
j
兲
册
⫹
4m
q
2
关
q
ជ
丢 q
ជ
兴
(2)
•
关
ជ
i
丢
ជ
j
兴
(2)
q
2
冎
, 共5兲
where the ’s are the color Gell-Mann matrices and
␣
s
is the
strong coupling constant.
For the present study we make use of the nonlocal NN
potential derived through a Lippmann-Schwinger formula-
tion of the RGM equations in momentum space 关18兴. The
formulation of the RGM for a system of two baryons B
1
and
B
2
needs the wave function of the two-baryon system con-
structed from the one-baryon wave functions. The two-
baryon wave function can be written as
⌿
B
1
B
2
⫽ A
关
共
P
ជ
兲
⌿
B
1
B
2
ST
兴
⫽ A
关
共
P
ជ
兲
B
1
共
p
ជ
B
1
兲
B
2
共
p
ជ
B
2
兲
B
1
B
2
ST
c
关
2
3
兴兴
, 共6兲
where A is the antisymmetrizer of the six-quark system,
(P
ជ
) is the relative wave function of the two clusters,
B
i
(p
ជ
B
i
) is the internal spatial wave function of the baryon
B
i
, and
B
i
are the internal coordinates of the three quarks of
baryon B
i
.
B
i
B
2
ST
denotes spin-isospin wave function of the
two-baryon system coupled to spin 共S兲 and isospin (T), and,
finally,
c
关
2
3
兴
is the product of two color singlets.
The dynamics of the system is governed by the Schro
¨
-
dinger equation
共
H⫺ E
T
兲
兩
⌿
典
⫽ 0⇒
具
␦
⌿
兩
共
H⫺ E
T
兲
兩
⌿
典
⫽ 0, 共7兲
where
TABLE I. Quark-model parameters. The values in brackets are
used for a correct description of the deuteron.
m
q
共MeV兲 313
b
a
共fm兲 0.518
␣
s
0.4977
g
ch
2
6.60 共6.86兲
m
S
(fm
⫺1
) 3.400
m
PS
(fm
⫺1
) 0.70
⌳
(fm
⫺1
) 4.47
a
b is the parameter of the harmonic oscillator wave function used
for each quark
(x)⫽(1/
b
2
)
(3/4)
e
⫺ (x
2
/2b
2
)
.
TABLE II. NN properties.
Quark model Nijm II 关22兴 Bonn B 关21兴 Expt.
Low-energy scattering parameters
1
S
0
a
s
共fm兲 ⫺ 23.759 ⫺23.739 ⫺ 23.750 ⫺ 23.74⫾ 0.02
r
s
共fm兲 2.68 2.67 2.71 2.77⫾ 0.05
3
S
1
a
t
共fm兲 5.461 5.418 5.424 5.419⫾ 0.007
r
t
共fm兲 1.820 1.753 1.761 1.753⫾ 0.008
Deuteron properties
⑀
d
共MeV兲 ⫺ 2.2242 ⫺ 2.2246 ⫺2.2246 ⫺ 2.224575
P
D
共%兲 4.85 5.64 4.99 ⫺
Q
d
(fm
2
) 0.276 0.271 0.278 0.2859⫾0.0003
A
S
(fm
⫺ 1/2
) 0.891 0.8845 0.8860 0.8846⫾0.0009
A
D
/A
S
0.0257 0.0252 0.0264 0.0256⫾ 0.0004
JULIA
´
-DI
´
AZ, HAIDENBAUER, VALCARCE, AND FERNA
´
NDEZ PHYSICAL REVIEW C 65 034001
034001-2
H⫽
兺
i⫽ 1
N
p
ជ
i
2
2m
q
⫹
兺
i⬍ j
V
ij
⫺ T
c.m.
, 共8兲
with T
c.m.
being the center-of-mass kinetic energy, V
ij
the
quark-quark interaction described above, and m
q
the con-
stituent quark mass.
Assuming the functional form
B
共
p
ជ
兲
⫽
冉
b
2
冊
3/4
e
⫺ b
2
p
2
/2
, 共9兲
where b is related to the size of the nucleon quark core, Eq.
共7兲 can be written in the following way, after the integration
of the internal cluster degrees of freedom:
冉
P
ជ
2
2
⫺ E
冊
共
P
ជ
兲
⫹
冕
关
V
D
共
P
ជ
,P
ជ
i
兲
⫹ W
L
共
P
ជ
,P
ជ
i
兲
兴
共
P
ជ
兲
dP
ជ
i
⫽ 0.
共10兲
V
D
(P
ជ
,P
ជ
i
) is the direct RGM kernel and W
L
(P
ជ
,P
ជ
i
) is the
exchange RGM kernel, composed of three different terms
W
L
共
P
ជ
,P
ជ
i
兲
⫽ T
L
共
P
ជ
,P
ជ
i
兲
⫹ V
L
共
P
ជ
,P
ជ
i
兲
⫹
共
E⫹ E
in
兲
K
L
共
P
ជ
,P
ជ
i
兲
,
共11兲
where E
in
is the internal energy of the two-body system,
T
L
(P
ជ
,P
ជ
i
) is the kinetic energy exchange kernel, V
L
(P
ជ
,P
ជ
i
)is
the potential energy exchange kernel, and K
L
(P
ជ
,P
ជ
i
) is the
exchange norm kernel. Note that if we do not mind how
V
D
(P
ជ
,P
ជ
i
) and W
L
(P
ជ
,P
ជ
i
) were derived microscopically, Eq.
共10兲 can be regarded as a general single-channel equation of
motion with including energy-dependent nonlocal potential.
V
D
(P
ជ
,P
ជ
i
), which contains the direct RGM potential, and
W
L
(P
ជ
⬘
,P
ជ
i
), which contains the exchange RGM potential
coming from quark antisymmetry, constitute our energy-
dependent nonlocal potential. In our case E
in
⫽ 2m
N
what
makes our potential almost energy independent, because the
center-of-mass energy of the two-body system, E, is much
smaller than the internal energy E
in
.
The potential yields a fairly good reproduction of the ex-
perimental data up to laboratory energies of 250 MeV. For a
correct description of the
1
S
0
phase shift it is necessary to
take into account the coupling to the
5
D
0
N⌬ channel 关17兴,
which provides an isospin-dependent mechanism generating
the additional attraction in this channel. This is implemented
in our calculation generalizing Eq. 共10兲 to a coupled channel
scheme. It implies a modification of Eq. 共11兲 with an addi-
tional term which contains the NN→N⌬ coupling. The pa-
rameters used are summarized in Table I. In Table II we
present the results for the low-energy scattering data and the
deuteron properties of the present model together with values
of some standard NN potentials 关21,22兴 and experimental
data. It is known that a charge symmetry breaking term
should be included in the interaction if one wants to repro-
duce those quantities simultaneously 关23兴. This is taken into
account by a slight modification of the chiral coupling con-
stant to reproduce the deuteron and the low-energy scattering
parameters 共see Table I兲. We also show, in Figs. 1, 2, and 3,
the
1
S
0
and
3
S
1
-
3
D
1
phase shifts and the mixing parameter
1
in comparison to results from phase-shift analyses 关24–
26兴.
FIG. 1.
1
S
0
NN phase shift. The solid line is the result for the
nonlocal quark-model potential. The dotted line shows the result of
the separable representation of the nonlocal quark-model potential.
The squares, diamonds, and triangles are the experimental data
taken from Refs. 关24兴, 关25兴, and 关26兴, respectively.
FIG. 2. Phase shifts for the
3
S
1
and
3
D
1
partial waves. Same
description as in Fig. 1 except for the dotted line which is not
shown.
TABLE III. Expansion 共lab兲 energies E
共in MeV兲 used in the EST representations of the quark-model
potential.
⑀
d
refers to the deuteron binding energy. l
is the boundary condition chosen for the angular
momentum l
of the initial state 关30,31兴.
Partial wave (E
,l
)
1
S
0
(NN)⫺
5
D
0
(N⌬) 共0,0兲共50,0兲共300,0兲共⫺20,0兲共⫺20,2兲共⫺50,0兲
3
S
1
-
3
D
1
⑀
d
共100,0兲共175,2兲共300,2兲共⫺50,0兲共⫺50,2兲
QUARK-MODEL BASED STUDY OF THE TRITON . . . PHYSICAL REVIEW C 65 034001
034001-3
III. TRITON BINDING ENERGY
The triton binding energy is obtained by means of a Fad-
deev calculation using the NN interaction described above.
We perform a so-called five-channel calculation; i.e., we use
only the
1
S
0
and
3
S
1
-
3
D
1
NN partial waves as input. Note
that since in our model there is a coupling to the N⌬ system,
as explained above, a fully consistent calculation would re-
quire the inclusion of two more three-body channels. How-
ever, their contribution to the 3N binding energy is known to
be rather small 关27兴 and therefore we neglect them for sim-
plicity reasons.
To solve the three-body Faddeev equations in momentum
space we first perform a separable finite-rank expansion of
the NN(⫺ N⌬) sector utilizing the Ernst-Shakin-Thaler
共EST兲 method 关28兴. Such a technique has been extensively
studied by one of the authors 共J.H.兲 for various realistic NN
potentials 关29兴 and specifically for a model that also includes
a coupling to the N⌬ system 关30兴. In those works it was
shown that, with a separable expansion of sufficiently high
rank, reliable and accurate results on the three-body level can
be achieved. In the present case it turned out that separable
representations of rank 6 — for
1
S
0
-(
5
D
0
) and for
3
S
1
-
3
D
1
— are sufficient to get converged results. The set of
energies used for the EST separable representations is listed
in Table III. We refer the reader to Refs. 关29–31兴 for techni-
cal details on the expansion method. The quality of the sepa-
rable expansion on the NN sector can be seen in Fig. 1,
where we show phase shifts for the original nonlocal poten-
tial and for the corresponding separable expansion. Evi-
dently, the phases are almost indistinguishable.
Results for the triton are summarized in Table IV. It is
reassuring to see that the predicted triton binding energy is
comparable to those obtained from conventional NN poten-
tials, such as the Bonn or Nijmegen models. Thus, our cal-
culations show that quark-model based NN interactions are
definitely able to provide a realistic description of the triton.
The results also give support to the use of such an interaction
model for further few-body calculations. One should not for-
get at this point that the number of free parameters is greatly
reduced in quark-model based NN interactions like the
present one. In addition, the parameters are strongly corre-
lated by the requirement to obtain a reasonable description of
the baryon spectrum.
IV. CONCLUSIONS
We have calculated the three-nucleon bound state problem
utilizing a nonlocal NN potential derived from a basic quark-
quark interaction. This potential was generated by means of
the resonating group method so that nonlocalities resulting
from the internal structure of the nucleon were preserved.
The resulting triton binding energy is comparable to those
obtained from conventional NN potentials.
In the calculation of the three-nucleon binding energy we
have followed the traditional approach: namely, solving the
Faddeev equations with nucleon degrees of freedom. Let us
remark, however, that in a more fundamental approach one
would impose consistency between the treatment of two- and
three-nucleon systems in terms of quark degrees of freedom.
That, of course, would require a derivation and solution of
the corresponding three-nucleon RGM equations. In such a
framework the quark structure of nucleons generates 共besides
the consecutive two-nucleon interactions that are summed up
by the Faddeev equations兲 also genuine three-body forces.
These forces are of short-ranged nature and they could be
significant for short-distance phenomena like the high-
momentum-transfer part of the charge form factor of
3
He.
Indeed, there have been attempts to explore the effects of
such three-body forces on the triton binding energy. In a
simple model based on a single one-gluon exchange 关32兴 the
three-body exchange kernels have been evaluated. An esti-
mation provided in this reference suggests that those three-
nucleon forces could yield additional binding in the order of
0.2 MeV. If this is the case, then those effects would be still
small enough to guarantee that the approach we followed in
our study is sufficiently accurate for an exploratory calcula-
tion. However, one has to keep in mind that the estimation in
Ref. 关32兴 was done only in perturbation theory and by means
of a zeroth-order three-nucleon wave function with a series
of fitted parameters. Thus, for the future, a more refined and
consistent treatment of the three-nucleon problem within the
quark picture is certainly desirable in order to allow for re-
liable conclusions on this issue.
ACKNOWLEDGMENTS
The authors thank D. R. Entem for providing the codes
used for the nonlocal interaction potential and for many use-
FIG. 3. Mixing parameter
1
. Same description as in Fig. 2.
TABLE IV. Properties of the three-nucleon bound state.
Quark model Nijm II Bonn B 关29兴
E
B
共MeV兲 ⫺7.72 ⫺7.65 ⫺8.17
P
S
共%兲 91.49 90.33 91.35
P
S
⬘
共%兲 1.430 1.339 1.368
P
P
共%兲 0.044 0.064 0.049
P
D
共%兲 7.033 8.267 7.235
JULIA
´
-DI
´
AZ, HAIDENBAUER, VALCARCE, AND FERNA
´
NDEZ PHYSICAL REVIEW C 65 034001
034001-4
ful comments. One of the authors 共B.J-D.兲 wants to thank the
hospitality of the FZ Ju
¨
lich where part of this work was
done. A.V. thanks the Ministerio de Educacio
´
n, Cultura y
Deporte of Spain for financial support through the Salvador
de Madariaga program. This work was partially funded by
Direccio
´
n General de Investigacio
´
n Cientı
´
fica y Te
´
cnica
共DGICYT兲 under Contract No. PB97-1401 and by Junta de
Castilla y Leo
´
n under Contract No. SA-109/01.
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