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\title{THREE-BODY CONFINEMENT FORCE IN A REALISTIC 
CONSTITUENT QUARK MODEL}
\author{Z.\ Papp\footnote{e-mail: Zoltan.Papp@atomki.hu}}
\address{Institute of Nuclear Research, H-4001 Debrecen, Pf. 51, 
Hungary}
\author{\underline {Fl.\ Stancu}\footnote{e-mail: fstancu@ulg.ac.be}}
\address{University of Li\`ege, Institute of Physics B.5, Sart Tilman,
B-4000 Li\`ege 1, Belgium}
\date{\today}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{minipage}{0.99\textwidth}

\begin{abstract}
We implement a three-body confinement force, introduced
on an algebraic basis, into
a semirelativistic version of the Goldstone-boson-exchange
constituent quark model. By solving the Faddeev equations
we show that this interaction can increase the  
gap between singlet and colour states, such as the latter can be ignored
and the known baryons can be described as $q^3$ systems.
%We present results for 
We analyze the effect of a $\Delta$- and a Y-shape three-body interaction.
%in terms of  
%the relative strength of the three-body versus the 
%usual two-body confinement interaction.
%lead to similar results.
\end{abstract}

%\pacs {PACS numbers:} 12.39.-x,12.39.Pn,12.40.Yx 
PACS: 12.39.-x,12.39.Pn,12.40.Yx

Keywords: Phenomenological quark models, SU(3) invariants,
three-body confinement forces, Faddeev approach.

\end{minipage}
\end{center}

\newpage
\vspace{1cm}
In constituent quark models
only two-body $F_i \cdot F_j$ colour confinement forces
have been considered so far.
They can be expressed in terms of the
quadratic (Casimir) operator of SU$_C$(3), thus the three-quark Hamiltonian 
of a constituent quark model is SU$_C$(3) invariant. But as such, 
besides colour singlets, the Hamiltonian also possesses
colour octet and decuplet states. In practice, these states have tacitly 
been ignored,
relying on the assumption that colour states do not exist.
However, if these states are located in the observed 
baryon resonance region
they should either be taken into account as 
coupled to colour octet sea $q \overline q$ pairs  
or gluons 
(see for example \cite{LIPKIN,CHENG_LI,LINDE})
or otherwise shifted at higher energies, by a change in the dynamics 
of the quark model. 

A solution in the latter direction 
has recently been proposed in Ref. \cite{DMI}.
Although OCD has a local exact SU$_{C}$(3) symmetry, in 
QCD-inspired models this symmetry appears as global, by construction.
%Based on the 
%argument that SU$_{C}$(3) is an exact global symmetry of QCD,
%which 
This implies that any quark model Hamiltonian inspired by QCD can be
written in terms of every SU(3) invariant operator.
Based on this argument in Ref. \cite{DMI}  a three-quark 
confining potential
that depends on the cubic invariant operator of SU(3) has 
been added to the usual two-body confinement. In a qualitative way, 
it was shown that such an interaction can 
increase the gap between the singlet and colour states provided
its strength has a specific sign and range.
Besides its implications on the spectrum of ordinary $q^3$ systems, its 
role in  exotic $q^2
\overline {q}^2$ hadrons has also been considered. 

The three-body confinement force of Ref.\ \cite{DMI} has  
subsequently been analysed in Ref.\ \cite{PS} in the context of a harmonic 
oscillator confinement where the size of its effect 
on the gap between the 
colour and singlet states in the spectrum of a $q^3$ system has 
been estimated. 
As a by-product, the $q^6$ system with relevance to 
the nucleon-nucleon problem has also been discussed.

The existence of a three-body confinement force has been previously
mentioned in the literature (see e.g. \cite{ROBSON}) as being
more complex than summing over the confining interaction between
pairs, but no particular form for its colour dependence
has been suggested. 

In the present work we first show explicitly how much the 
singlet and colour states of a realistic constituent quark model 
appear close to each other. Next, by introducing a three-body
confinement force related to the cubic invariant of SU(3), 
we demonstrate that one can separate them largely enough so that the 
colour states can be ignored. For this purpose, here   
we consider the semirelativistic 
constituent quark model of Ref.\ \cite{FRASCATI},
into which we implement a three-body confinement interaction. 
By using the Faddeev approach of Ref.\ \cite{PAPP} we calculate the lowest singlet, octet and 
decuplet states
of a three-quark system, 
with this three-body confinement force. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Without three-body confinement we find that the lowest colour octet state 
%with S=1/2 and I=1/2
%is located around 1700 MeV, i.\ e.\ in the low-energy part of the  
%experimentally known nucleon spectrum and the
%colour decuplet, with the same spin and isospin, is located at 
%about 2600 MeV, i.\ e.\ in the high-energy part of the 
%experimentally known nucleon spectrum. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Here we present the nucleon sector, namely S=1/2 and I=1/2. 
The lowest colour octet and decuplet $q^3$ states compatible 
with the Pauli principle are displayed in Tables \ref{octet}
and \ref{decuplet} respectively .
For orientation, their excitation energy $E_N-E_0$,
estimated in a harmonic oscillator confinement,
is shown in the last column.
In the same harmonic oscillator confinement the first colourless  
negative parity state is also 616 MeV above the ground state.
Therefore one can see that the $0^+$ colour octet and decuplet states 
appear substantially below this negative parity state.
In the following we shall consider only the lowest colour states,
namely the $0^+$ and $1^-$ octets and the  $0^+$ decuplet.
It is necessary
to displace all colour states at energies where they can safely be neglected,
if the nucleon resonances are to be interpreted as pure $q^3$ excitations.
This can indeed be achieved by an
additional three-body force, as shown below. 
 
The realistic total Hamiltonian presently under discussion  
has the form
\begin{equation}\label{HAMILT}
H = \sum\limits_{i}^{} \left(m_i^2 + p_i^2\right)^{1/2}
 + V_{2b} + V_{3b} + V_{\chi}~,
\end{equation}
where $\vec p_i$ and $m_i$ are the momentum and the constituent mass of the
quark $i$,
$V_{2b}$ the two-body confinement, 
$V_{3b}$ the three-body confinement and $V_{\chi}$ the hyperfine
interaction.  
Taking $V_{3b}$ = 0 one recovers the chiral constituent quark model 
of Ref.\ \cite{FRASCATI} which is the simplest semirelativistic version
(no tensor, no spin-orbit)
of a model originally proposed in Ref.\ \cite{GR96} where the 
hyperfine interaction 
is spin and flavour dependent. 
The origin of such an interaction is thought to be the pseudoscalar meson 
exchange between quarks. This model,
either in a nonrelativistic or a semirelativistic form,
reproduces well the spectra of light baryons
and in particular the correct order of positive and negative parity 
states. Although we have chosen a particular model  
we expect our conclusions to be relevant  
for any reasonable constituent quark model with three valence quarks.
The detailed parametrization of $V_{\chi}$ can be found in Ref.\ 
\cite{FRASCATI} and we do not reproduce it here, the main emphasis
being laid on the confinement potential. We use 
a $V_{2b}$ term of the form
\begin{equation}\label{V2b}
V_{2b} = \sum\limits_{i<j} {\em v}^c(r_{ij})~(\frac{7}{3} 
+ F^{a}_{i} F^{a}_{j})~,
\end{equation}
where $F^{a}_{i} = \frac{1}{2} \lambda^{a}_{i} $ is the colour charge
operator of the quark $i$, $\lambda^a$ are the Gell-Mann matrices 
and 
\begin{equation}\label{RADIAL}
{\em v}^c (r_{ij}) = V_0 + \gamma r_{ij}~, 
\end{equation}
with $\gamma=2.33$ fm$^{-2}$
and $V_0= -416$ MeV as in Ref.\ \cite{FRASCATI}. The colour part
of $V_{2b}$ is 
consistent with Refs.\ \cite{DMI,PS} but different from that of \cite{FRASCATI}.
It stems from the stability condition of a $q \overline q$ pair
but it depends on an arbitrary constant which leads to the term $\frac{7}{3}$.
Anyhow, we rescale this interaction such as to reproduce the spectrum
obtained in \cite{FRASCATI} (see later).

The three-body confinement interaction has the form \cite{DMI} 
\begin{equation}\label{3BTOTAL}
V_{3b}=V_{ijk} = {\mathcal V}_{ijk} {\mathcal C}_{ijk}~,
\end{equation}
where ${\mathcal V}_{ijk}$ is the radial part and ${\mathcal C}_{ijk}$
is the colour operator
\begin{equation}\label{3B}
{\mathcal C}_{ijk} = d^{abc}~F^{a}_{i} ~F^{b}_{j}~ F^{c}_{k}~,
\end{equation}
with $d^{abc}$ some real constants,
symmetric under any permutation of indices and
defined by the anticommutator of the Gell-Mann matrices  as
$\{ \lambda^a,  \lambda^b \} = 2 d^{abc}~\lambda^{c}$ \cite{book}.
The operator (\ref{3B}) can be rewritten in terms of the two
independent invariant operators of SU(3) as 
\begin{equation}\label{EQ4}
{\mathcal C}_{ijk} =
%d^{abc}~F^{a}_{i} ~F^{b}_{j}~ F^{c}_{k} =
\frac{1}{6}~ [~  C^{(3)}_{i+j+k} - \frac{5}{2} C^{(2)}_{i+j+k} +
\frac{20}{3}~ ]~,
\end{equation}
where $C^{(2)}$ is the quadratic (Casimir) and $C^{(3)}$
the cubic invariant. The expectation values of (\ref{EQ4})
can thus easily be obtained from the eigenvalue of $C^{(2)}$ and
$C^{(3)}$ \cite{book} for any irreducible representation of SU(3).

The choice of ${\mathcal V}_{ijk}$ is related to our present 
knowledge of confinement.  
Lattice calculations are ambiguous about its form in baryons.
Both Y and $\Delta$ shapes are supported 
\cite{BALI}. Moreover
the interaction potential obtained in such calculations corresponds
to the colourless ground state only. No information from lattice
QCD about colour octets is available so far.
In this situation we use a three-body confinement, either in a 
triangular shape 
\begin{equation}\label{DELTA}
 {\mathcal V}_{ijk} = {\gamma} ~c~[|\vec{r_i}-\vec{r_j}|
+|\vec{r_j}-\vec{r_k}|+|\vec{r_k}-\vec{r_i}|]~,
\end{equation}
or in a Y-shape  
\begin{equation}\label{Y}
 {\mathcal V}_{ijk} = {\gamma}~c~[|\vec{r_i}-\vec{r_0}|
+|\vec{r_j}-\vec{r_0}|+|\vec{r_k}-\vec{r_0}|]~,
\end{equation}
with $\vec{r}_0$ the point
where the three flux tubes meet such as to satisfy the SU$_{C}$(3) gauge
invariance (see e.\ g.\ \cite{CKP}).
In (\ref{DELTA}) and (\ref{Y})
the value of $\gamma$ is the same as in Eq.\ (\ref{RADIAL})
and the parameter $c$ represents the relative strength of the 
three-body versus the two-body force. By assuming a triangular shape,
in Ref.\ \cite{DMI} it was found that $c$ must be located in the interval  
$-\frac{3}{2}~<~c~<~\frac{2}{5}.$
The upper limit ensures that the lowest
colour singlet is below the lowest colour octet state. The lower limit was   
required by the stability condition of the nucleon, 
$\langle V_{2b} + V_{3b} \rangle > 0$. 
In Ref.\ \cite{PS} some arbitrariness was noticed for 
the lower bound because this is related to the choice of
the colour operator in (\ref{V2b}). 
However the above range of $c$
is entirely satisfactory for our discussion of the triangular shape.
In fact the common conclusion of 
\cite{DMI} and \cite{PS} was that $c$ must be negative in order
to obtain an increase in the gap between the colour octet and singlet states
in a $q^3$ system.
Therefore 
in the calculations below related to the triangular shape, we take 
\begin{equation}\label{INEQ1}
-\frac{3}{2}~<~c~<~0~.
\end{equation}

The Hamiltonian (\ref{HAMILT}) was solved by using the Faddeev approach of
Ref.\ \cite{PAPP}, adequate for confining potentials. 
The necessary expectation values for the two-body colour
operator appearing in (\ref{V2b}) are given in Table \ref{twobody}. The
expectation values of the three-body colour operator (\ref{EQ4}) are taken
from \cite{DMI} or \cite{PS}. These are 10/9, -5/36 and 1/9 for
the singlet, octet and decuplet SU(3) states, respectively. 
%Here we present results for states belonging to the S=1/2, I=1/2 sector,
%including the ground state nucleon.  

A particular advantage of the Faddeev approach is that the incorporation
of permutation symmetry is very easy. For identical quarks, as in this
case, the three Faddeev components of the three-quark
wave function have the same functional form in their 
own Jacobi coordinate systems.
Therefore the three equations can be reduced to a single one. From the 
structure  
of this equation it follows that the correct symmetry of the wave function under
the exchange of any two particles is automatically guaranteed if the correct
symmetry is implemented in one of the three components.
In our calculation a bipolar harmonic basis was used,
combined with spin, isospin and colour basis states.
If we select the basis states such as
\begin{equation} \label{permut}
(-)^{l + s + i + c} = - 1~,
\end{equation}
where $l$, $s$, $i$ and $c$ are the relative angular momentum, spin, isospin
and colour quantum numbers of a two-quark pair, the Pauli principle for the 
three-quark system is satisfied.
We have $(-)^{c}$ = 1 when $[\tilde{f}]_C=[2]$ and $(-)^{c}$ = - 1 when 
$[\tilde{f}]_C=[11]$, where $[\tilde{f}]_C$ is a given partition.
If we denote the total angular momentum  by $L$ and the 
total parity by $P = (-)^{l + \lambda}$, where $\lambda$ is the relative
angular momentum of the third particle with respect to the pair,
then the lowest colour octets must have $L^P = 0^+, 1^{-1}$ 
and the lowest decuplet $L^P = 0^+$.
In Table \ref{pauli} we 
show the structure of 
the lowest colour states with the corresponding 
quantum numbers of their components. 
This treatment of permutation symmetry as well as the whole numerical procedure
were checked against the exact harmonic oscillator results of Ref.\ \cite{PS}.
%Note however that in Ref.\ \cite{PS} only states of 
%spin-isospin symmetry $[3]$, i.e. the same as that of the nucleon, have been
%considered. Here we enlarge the spin-isospin symmetry space
%to incorporate the $[21]$ symmetry as well.

As mentioned above, our purpose is to understand the implications of 
a three-body colour confinement interaction in a realistic model.  We look
separately at its effect produced either by the $\Delta$ shape (\ref{DELTA}),
or by the Y shape (\ref{Y}).  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\vspace{0.5cm}
{\bf I. The $\Delta$-shape.} This shape is currently used in conjunction
with a two-body $F_i \cdot F_j$ colour operator as an approximation
to the Y shape  
\cite{CKP}. In this case the contribution of the colour part of 
$V_{2b}$ and $V_{3b}$ sum up together to $\chi_i$ (i=1,8,10)
as in Eq.\ (14) of Ref.\ \cite{PS}.
In particular the expectation value 
of the colour operator contained in $V_{2b} + V_{3b}$ is
$\chi_1$ = 5/3 + 10/9 $c$ for the singlet.
In order to keep the ground state baryon
mass constant and equal to that of Ref.\ \cite{FRASCATI} 
we rescaled $\chi_i$ to
\begin{equation}\label{CHI}
\overline {\chi_i} = \left\{ \renewcommand{\arraystretch}{2}
\begin{array}{cl}
 1  &\hspace{1.5cm} \mbox{ i=1 (singlet)} \\
 (\frac{13}{6} - \frac{5}{36} c)/(\frac{5}{3} + \frac{10}{9} c)
 & \hspace{1.5cm} \mbox{ i=8 (octet)} \\
 (\frac{8}{3} + \frac{1}{9} c)/(\frac{5}{3} + \frac{10}{9} c)
  & \hspace{1.5cm} \mbox{ i=10 (decuplet)}
\end{array} \right.
\label{colour}
\end{equation}

In Fig.\ 1 we show the dependence of some eigenvalues of the
Hamiltonian (\ref{HAMILT}) as a function of $-c$ for $-1.4~ \leq c~ \leq 0$.
The case $c$ = 0 is the model \cite{FRASCATI} for
which we reproduced the ground state nucleon mass $m_N$=940 MeV, 
and the resonance masses N(1440 $\frac{1}{2}^{+})$= 1459 MeV,
N(1535 $\frac{1}{2}^{-})$ = 1522 MeV 
and N(1710 $\frac{1}{2}^{+})$= 1783 MeV. With the confinement
strength (\ref{CHI}) the whole baryon spectrum, including the
above states, is independent of $c$, as expected. For $c=0$
the lowest colour octets $0^+$ and $1^{-}$ 
acquire the masses 1536 MeV and 
1758 MeV respectively and the $0^+$ colour decuplet has 2077 MeV. 
Thus in a 
constituent quark model with three valence quarks the colour states
are so low that they cannot be ignored. If the three-body interaction
(\ref{DELTA}) is switched on, the gap between the lowest colour 
singlet S=1/2, I=1/2 states and the colour $q^3$ states increases. For 
$c~ < -1.2$ they are enough far apart (for example the $0^+$ colour octet    
raises to 8852 MeV  when $c= -1.4$ ) and thus can 
safely be neglected.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

{\bf II. The Y-shape.} This is a genuine three-body force, both in the
coordinate and the colour space as well.  
Accordingly, the addition low (\ref{CHI}) does not hold anymore.
In the Y-shape, the flux tubes meet at 120$^0$ in order to ensure the 
minimum energy. This shape moves continuously to a two-legged flux-tube
configuration where the legs meet at an angle larger than 120$^0$. In this
case one of the flux tubes collapses which means that 
$\vec{r}_i$= $\vec{r}_0$ for $i$ = 1, 2 or 3.  
The spectrum associated to the Y-shape (\ref{Y}) is presented in 
Fig.~\ref{Fig2}. In this case the lower limit on $c$ 
imposed by the inequality (\ref{INEQ1}) is no more valid but $c$  
must remain negative. So we varied $c$ between zero and - 2.
At $c = 0$ the spectrum is the same as in Fig.~\ref{Fig1}. 
In  the interval $-2 \leq c \leq 0$ 
the resonance N(1710 $\frac{1}{2}^{+})$ rises
by about 50 MeV,  the other colour singlet 
states remaining practically unaffected by the presence of the three-body 
Y-shape force. 
For $ c < - 2$ the colour singlet spectrum is more substantially affected and
a reparametrization of the model \cite{FRASCATI} would be
required. Anyhow, around $c = - 2$ the colour states are far 
away from the 
presently observed nucleon spectrum and can be neglected. 
The trend is similar to the $\Delta$-shape case, but the mass of any
colour state rises  slower with the strength 
$|c|$ of the three-body Y-shape force. 
If one would rescale the $|c|$-axis
by a factor 1/2,
% resulting from the SU(3) algebra and already
%contained in the parameter $\gamma$ of Ref. \cite{FRASCATI}, 
the $\Delta$- and the Y-shape results  
would look closer to each other. 


   To better understand the role of the three-body confinement force we 
also calculated the root-mean-square radii associated to the states
of Fig.~\ref{Fig2}. These are displayed in Fig.~\ref{Fig3}
except for the radius of the $1^-$ state, which is practically identical
to that of the $0^+$ state. 
One can see that the quark core 
radii of the ground state nucleon and its lowest 
N($\frac{1}{2})$ 
resonances remain practically unchanged. On the other hand, the colour
states shrink substantially, becoming 
smaller than the nucleon itself beyond $c < - 1$. This 
is obviously the effect 
of the increase of the confinement contribution through the
addition of the three-body term and may be an welcome feature.


\vspace{1cm}

%{\bf Conclusions.}
In conclusion, through the example of the Goldstone-boson-exchange 
model\ \cite{FRASCATI} 
we have shown that the spectrum of a realistic constituent quark 
model with a pairwise colour  confinement operator
and a linearly increasing behaviour
can accomodate both singlet and colour low-lying $q^3$ states.
The lowest octet and decuplet
colour states appear in the middle of the observed spectrum
which means they cannot be ignored. In this situation there are
two alternatives:

1) To keep simplicity. Then, as shown here,
the addition of a three-body confinement interaction
can increase the gap between 
singlet and colour states by 
a few GeV without altering the quality of the 
colour singlet spectrum. Then the colour states can be neglected
in calculations and the baryons can be described as pure $q^3$ states.
The size of the gap depends on the spatial
part of the three-body force. The relative strength $c$ of the
three-body force versus the two-body confinement force must 
be stronger for the Y-shape than for the more common $\Delta$-shape
in order to obtain a similar energy separation.
It would be interesting
to see how other baryon properties are affected by a 
three-body confinement interaction.


2) If however the colour states are maintained low in the spectrum
of a Hamiltonian with a pairwise confinement interaction only,
one has to give up the simple $q^3$ picture of baryons. The 
colour states could give rise to colourless hybrid baryons having 
singlet-singlet + octet-octet colour components either of type 
$(qqq)(q \overline q)$  or  of type $(qqq)g$  
\cite{LIPKIN,CHENG_LI,LINDE}. In the context of a potential
model based on a one-gluon exchange hyperfine interaction, 
applied to the study of tetraquarks i. e. $q \overline q q \overline q$ systems, 
it has been shown \cite{WEINSTEIN,JMR,BRINK} that there is a
strong mixing between  states formed of two colour singlet and two colour octet
$(q \overline q)$ pairs which leads to a substantial lowering of the
variational energy.
This should also be the case for hybrid baryons
where singlet-singlet $ (qqq)_1 (q \overline q)_1$ and 
octet-octet $ (qqq)_8 (q \overline q)_8$ components could mix
substantially. The present study suggests that such hybrids are to be expected 
in all partial waves if only two-body $F_i \cdot F_j$ are considered.
This deserves a separate investigation
inasmuch as hybrid baryons
raise a new interest \cite{PAGE}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace{1cm}
{\it Acknowledgments} We are most grateful to Stephane Pepin and 
Jean-Marc Richard for
useful suggestions and a careful reading of the manuscript. One of us 
(Z.P.) acknowledges
financial support from the {\it Fond National de la Recherche
Scientifique} of Belgium and is grateful for warm hospitality
at the Theoretical Fundamental Physics Laboratory of the
University of Li\`ege. This work has been partially supported
by OTKA Grants T026233 and T029003.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{references}

\bibitem{LIPKIN} H. J. Lipkin, Phys. Lett {\bf B251} (1990) 613.

\bibitem{CHENG_LI} T. P. Cheng and Ling-Fong Li, Phys. Rev. Lett. {\bf 74}
(1995) 2872.

\bibitem{LINDE} 
J. Linde, T. Ohlsson and H. Snellman, Phys. Rev. {\bf D57} 
(1998) 452.

\bibitem{DMI} V. Dmitrasinovic, Phys. Lett {\bf B499} (2001) 135.

\bibitem{PS} S. Pepin and Fl. Stancu, Phys. Rev. {\bf D65} (2002) 054032.

\bibitem{ROBSON} D. P. Stanley and D. Robson, Phys. Rev. Lett {\bf 45}
(1980) 235.
 
\bibitem{FRASCATI} L. Ya. Glozman, Z. Papp, W. Plessas, K. Varga and
R. F. Wagenbrunn, Phys. Rev. {\bf C57} (1998) 346; 
L. Ya. Glozman, W. Plessas, K. Varga and
R. F. Wagenbrunn, Nucl. Phys. {\bf A631} (1998) 469c; Phys. Rev.
{\bf D58} (1998) 094030. 

\bibitem{PAPP} Z. Papp, A. Krassnigg and W. Plessas,
Phys. Rev. {\bf C62} (2000) 044004.

\bibitem{GR96} L.Ya. Glozman and D.O. Riska, Phys. Rep. 268 (1996) 263.

\bibitem{book} Fl. Stancu, {\it Group Theory in Subnuclear Physics},
(Oxford University Press, Oxford, 1996).

\bibitem{BALI} G. Bali, Phys. Rep. {\bf 343} (2001) 1;
%\bibitem{TAKAHASHI}
T. T. Takahashi, H. Matsufuru, Y. Nemoto
and H. Suganuma, Phys. Rev. Lett. {\bf 86} (2001) 18;
%\bibitem{ALEXANDROU} 
C. Alexandrou, Ph. de Forcrand and A. Tsapalis,
Phys. Rev. {\bf D65} (2002) 054503.

\bibitem{CKP} J. Carlson, J. Kogut and V. R. Pandharipande,
Phys. Rev. {\bf D27} (1983) 233.

\bibitem{WEINSTEIN} J. Weinstein and N. Isgur, Phys. Rev. {\bf D27}
(1983) 588; ibid {\bf D41} (1990) 2236.

\bibitem{JMR} S. Zouzou, B. Silvestre-Brac, C. Gignoux and 
J. -M. Richard, Z. Phys {\bf C30} (1986) 457.

\bibitem{BRINK} D. M. Brink and Fl. Stancu, Phys. Rev. {\bf D49}
(1994) 4665.

\bibitem{PAGE} P. R. Page, arXiv:.
\end{references}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  Tables
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}
\renewcommand{\arraystretch}{1.5}
\parbox{18cm}{\caption[octet]{\label{octet} Three-quark colour octet states
$[21]_C$ 
compatible with the Pauli principle. The first column gives the orbital
angular momentum $L$ and parity $P$, the second, third and fourth columns
give the permutation symmetry in the orbital, spin-isospin ad 
spin-isospin-colour spaces respectively and the last column gives
an estimates of the excitation energy 
$E_N-E_0=(N+3) \hbar \omega [\overline \chi_8]^{1/2}$ of each state for a 
harmonic oscillator two-body confinement with $N$ quanta,
$\omega/c$ = 2 $fm^{-1}$ and $\overline \chi_8$ ($c$=0) given by 
Eq. (\ref{CHI}). }} 
\begin{tabular}{cccccc}
$ L^P$ &  $[f]_O$ & $[f]_{IS}$ & $[f]_{ISC}$ & N & $E_N-E_0$\\
       &        &            &             &     &  (MeV)     \\      
\tableline
$0^+$ & $[3]$    & $[21]$     & $[1^3]$     & 0 & 166 \\
$1^-$ & $[21]$   & $[3]$      & $[21]$      & 1 & 616 \\
$1^-$ & $[21]$   & $[21]$     & $[21]$      & 1 & 616 \\
$1^-$ & $[21]$   & $[1^3]$    & $[21]$      & 1 & 616 \\
$1^+$ & $[1^3]$   & $[21]$     & $[3]$       & 2 & 1066 \\
\end{tabular}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}
\renewcommand{\arraystretch}{1.5}
\parbox{18cm}{\caption[octet]{\label{decuplet} Same as 
Table \ref{octet} but for colour decuplet $[3]_C$ states
and $E_N-E_0=(N+3) \hbar \omega [\overline \chi_{10}]^{1/2}$ 
where $\overline \chi_{10}$ ($c$=0) comes from Eq. (\ref{CHI}) also. }}
\begin{tabular}{cccccc}
$ L^P$ &  $[f]_O$ & $[f]_{IS}$ & $[f]_{ISC}$ & N & $E_N-E_0$\\
       &        &            &             &     &  (MeV)     \\      
\tableline
$0^+$ & $[3]$    & $[1^3]$     & $[1^3]$     & 0  & 314 \\
$1^-$ & $[21]$   & $[21]$      & $[21]$      & 1  & 814 \\
$1^+$ & $[1^3]$  & $[3]$       & $[3]$       & 2  & 1312\\
\end{tabular}
\end{table} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\begin{table}
\renewcommand{\arraystretch}{1.5}
\parbox{18cm}{\caption[twobody]{\label{twobody} The expectation values 
of the two-body colour operator $O_{ij}=\frac{7}{3} + F_i \cdot F_j$ 
of Eq. (\ref{V2b}) $(i<j)$ between three-quark states 
$|(c_2 c_3) [\tilde f]_C; c_1 [f]_C\rangle $
required in the Faddeev calculations. The particles 2 and 3 are first
coupled to a symmetric $[2]$ or an antisymmetric $[1^2]$ state and
next to particle 1 to a total colour symmetry $[f]_C$.}}
\begin{tabular}{cccc}
$[\tilde f]_C$ & $[f]_C$ & Color operator & Expectation value \\
\tableline
$[2]$  & $[3]$   & $O_{ij}$ & 8/3 \\
$[2]$  & $[21]$  & $O_{23}$ & 8/3 \\
$[2]$  & $[21]$  & $O_{12}$ & 23/12 \\
$[2]$  & $[21]$  & $O_{13}$ & 23/12 \\
$[11]$ & $[21]$  & $O_{23}$ & 5/3 \\
$[11]$ & $[21]$  & $O_{12}$ & 29/12 \\
$[11]$ & $[21]$  & $O_{13}$ & 29/12 \\
$[11]$ & $[1^3]$ & $O_{ij}$ & 5/3 \\
\end{tabular}
\end{table} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}
\renewcommand{\arraystretch}{0.8}
\parbox{18cm}{\caption[pauli]{\label{pauli} The quantum numbers 
of the components of a $q^3$ totally antisymmetric state
of colour symmetry $[f]_C$, total angular momentum $L$, total spin S=1/2,
and total isospin I=1/2, compatible with the Pauli principle, 
as used in the Faddeev calculations.
Here $l$, $s$, $i$ and $[\tilde f]_C$ are the relative angular momentum, 
spin, isospin
and colour quantum numbers of a two-quark pair and $\lambda$ is the relative
angular momentum of the third particle with respect to the pair.
}}  
\begin{tabular}{ccccccc}
$[f]_C$  & $L$ & $l$ & $\lambda$ & $s$ & $i$ & $[\tilde f]_C$ \\
\tableline
$[21]$ &   0 &   0   &   0   &   0   &   0   & $[11]$  \\
       &     &   0   &   0   &   1   &   1   & $[11]$  \\
       &     &   1   &   1   &   1   &   0   & $[11]$  \\
       &     &   1   &   1   &   0   &   1   & $[11]$  \\
       &     &   0   &   0   &   1   &   0   & $[2]$   \\
       &     &   0   &   0   &   0   &   1   & $[2]$   \\
       &     &   1   &   1   &   0   &   0   & $[2]$   \\
       &     &   1   &   1   &   1   &   1   & $[2]$   \\ 
\tableline
$[21]$ &   1 &   0   &   1   &   0   &   0   & $[11]$  \\
       &     &   0   &   1   &   1   &   1   & $[11]$  \\
       &     &   1   &   0   &   0   &   1   & $[11]$  \\
       &     &   1   &   0   &   1   &   0   & $[11]$  \\
       &     &   1   &   0   &   0   &   0   & $[2]$  \\
       &     &   1   &   0   &   1   &   1   & $[2]$  \\
       &     &   0   &   1   &   0   &   1   & $[2]$  \\
       &     &   0   &   1   &   1   &   0   & $[2]$  \\  
\tableline
$[3]$  &   0 &   0   &   0   &   1   &   0   & $[2]$  \\
       &     &   0   &   0   &   0   &   1   & $[2]$  \\
       &     &   1   &   1   &   0   &   0   & $[2]$  \\  
       &     &   1   &   1   &   1   &   1   & $[2]$  \\
\end{tabular}
\end{table}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%   FIGURES
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
%\input{fig2.tex}
\centerline{\epsfxsize=14cm\epsfbox{n3d.eps}}
\caption{\label{Fig1}
S=1/2, I=1/2 eigenvalues 
of the Hamiltonian (\ref{HAMILT}) with the $\Delta$-shape
three-body interaction  (\ref{DELTA})
as a function of $-c$. The solid 
line is the ground state nucleon, the dotted line the N(1440) 
resonance (Roper), the dashed line the lowest negative parity
state N(1535), the dot-dashed line 
the N(1710) resonance.
The colour octets $0^+$ and $1^{-}$ are represented by 
diamonds and circles respectively and the $0^+$ colour
decuplet by squares.}
\end{figure}
%%%
\begin{figure}
\centerline{\epsfxsize=16cm\epsfbox{n3y.eps}}
\caption{\label{Fig2}
Same as Fig.~\ref{Fig1} but for the Y-shape interaction
(\ref{Y}). }
\end{figure}
%%%
\begin{figure}
\centerline{\epsfxsize=16cm\epsfbox{r2y.eps}}
\caption{\label{Fig3}
Root mean square radii of states of  Fig.~\ref{Fig2}.}
\end{figure}


\end{document}





