A Faddeev Calculation for Pentaquark Θ+ in Diquark Picture
with Nambu-Jona-Lasinio Type Interaction
H. Mineo1,2,∗), J.A. Tjon1,3,∗∗), K. Tsushima4,5,∗∗∗), and Shin Nan Yang1,†)
1 Department of Physics, National Taiwan University, Taipei 10617, Taiwan
2 Institute of Atomic and Molecular Sciences, Academia Sinica,
P.O.Box 23-166, Taipei 10617, Taiwan
3KVI, University of Groningen, The Netherlands
4National Center for Theoretical Sciences, National Taiwan University,
Taipei 10617, Taiwan
5Grupo de F́ısica Nuclear and IUFFyM, Universidad de Salamanca,
E-37008 Salamanca, Spain
A Bethe-Salpeter-Faddeev (BSF) calculation is performed for the pentaquark Θ+ in the
diquark picture of Jaffe and Wilczek in which Θ+ is a diquark-diquark-s̄ three-body system.
Nambu-Jona-Lasinio (NJL) model is used to calculate the lowest order diagrams in the two-
body scatterings of s̄D and DD. With the use of coupling constants determined from the
meson sector, we find that s̄D interaction is attractive while DD interaction is repulsive.
However, we can not find a bound 1
pentaquark state. Instead, a bound pentaquark Θ+
channel is obtained with unphysically strong vector mesonic coupling constants.
§1. Introduction
Experimental situation on the existence of Θ+ is currently filled with conflicting
positive and negative evidence and the question is still not yet fully settled.1)
Even if it turns out that Θ+ does not exist, it would still be theoretically interest-
ing to understand the absence of such an exotic. Many theoretical approaches have
been used, including quark models, QCD sum rules, and lattice QCD, in addition to
the chiral soliton model, to understand the properties and structure of Θ+ since its
first sighting.2) One of the most intriguing theoretical ideas suggested for Θ+ is the
diquark picture of Jaffe and Wilczek (JW)3) in which Θ+ is considered as a three-
body system consisted of two scalar, isoscalar, color 3̄ diquarks (D’s) and a strange
antiquark (s̄). It is based, in part, on group theoretical consideration. It would hence
be desirable to examine such a scheme from a more dynamical perspective.
It is known that diquark arises naturally from Nambu-Jona-Lasinio (NJL) model,
an effective quark theory in the low energy region.4) NJL model conveniently incor-
porates chiral symmetry and its spontaneously breaking which dictates the hadronic
physics at low energy. Models based on NJL type of Lagrangians have been very
successful in describing the low energy meson physics. Based on relativistic Fad-
∗) e-mail address: mineo@gate.sinica.edu.tw
∗∗) e-mail address: J.A.Tjon@phys.uu.nl
∗∗∗) e-mail address: tsushima@usal.es
†) e-mail address: snyang@phys.ntu.edu.tw
typeset using PTPTEX.cls 〈Ver.0.9〉
http://arxiv.org/abs/0704.1072v2
2 H. Mineo, J.A. Tjon, K. Tsushima, and S.N. Yang
deev equation, the NJL model has also been applied to the baryon systems.5), 6)
It has been shown that, using the quark-diquark approximation, one can explain
the nucleon static properties and the qualitative features of the empirical valence
quark distribution reasonably well.6) Consequently, we will employ NJL model to
describe the dynamics of a (s̄DD) three-particle system in Faddeev formalism. We
use relativistic equations to describe both the three-particle and its two-particle sub-
systems, namely, the Bethe-Salpeter-Faddeev (BSF) equation7) and Bethe-Salpeter
(BS) equations. In practice, Blankenbecler-Sugar reduction scheme is used to reduce
the four-dimensional integral equation into three-dimensional ones.
§2. SU(3)f NJL model and the diquark
The SU(3)f NJL model is a chirally symmetric four-fermi contact interaction
Lagrangian. With the use of Fierz transformations, the original NJL interaction
Lagrangian LI can be rewritten, for the qq̄ channel, as
LI,qq̄ = G1
(ψ̄λafψ)
2 − (ψ̄γ5λafψ)
(ψ̄γµλafψ)
2 + (ψ̄γµγ5λafψ)
(ψ̄γµλ0fψ)
2 + (ψ̄γµγ5λ0fψ)
(ψ̄γµλ0fψ)
2 − (ψ̄γµγ5λ0fψ)
+ · · · , (2.1)
where a = 0 ∼ 8, and λ0
I. For later use, we define G5 = G2 +
Gv , with
Gv ≡ G3 + G4. For the scalar, isoscalar diquark channel, interaction Lagrangian is
given by
LI,s = Gs
ψ̄(γ5C)λ2fβ
ψT (C−1γ5)λ2fβ
, (2.2)
where βAc =
λA(A = 2, 5, 7) corresponds to one of the color 3̄c states. C = iγ
is the charge conjugation operator, and λ′s are the Gell-Mann matrices.
The constituent quark and diquark masses can be obtained from the gap equation
and t-matrix of the diquark. Since we are only interested in a qualitatively study of
the interactions inside Θ+, we will use the empirical values of the constituent quark
masses Mu,d = 400 MeV, Ms = 600 MeV, and the diquark mass MD = 600 MeV as
obtained in Ref.8)
§3. Two-body interactions for s̄D and DD channels
In the JW model for Θ+,3) symmetry consideration requires that the the spatial
wave function of the two scalar-isoscalar, color 3̄ diquarks must be antisymmetric
and the lowest possible state is p-state. Since Θ+ is of JP = 1
, s̄ would be in
relative s-wave to the DD pair. Accordingly, we will consider only the configuration
where s̄D and DD are in relative s- and p-waves, respectively.
Fig. 1 shows the lowest order diagram, i.e., first order in LI,qq̄ in s̄D scat-
tering. Trace properties in Dirac and flavor space limit the vertex Γ to only the
vector-isoscalar term, (ψ̄γµλ0
ψ)2. For the DD interaction, the quark rearrangement
Faddeev Calculation for Pentaquark Θ+ 3
diagram gives no contribution because of its color structure. The lowest order non-
vanishing diagram of the first order in LI,qq̄ is given in Fig. 2 where only the direct
term is shown. The corresponding exchange diagram vanishes again because of the
color structure.
Γ=λ f
aγµ (a=0,8)
Fig. 1. s̄D potential of the lowest order
in LI,qq̄ .
Fig. 2. Lowest order diagrams in DD
scattering.
With the use of the interaction Lagrangians of Eqs. (2.1-2.2), we obtain the
following driving terms of Fig. 1 and 2, in the BS equations for s̄D and DD two-
particle systems,
< s̄fDf |V |s̄iDi > = (−v̄(ps̄i))(−iVs̄D)(pDi, pDf )v(ps̄f ),
Vs̄D =
GvFv(q
2)Ṽs̄D(pDi, pDf ), Ṽs̄D(pDi, pDf ) = (6pDi + 6pDf )/2 (3.1)
− iVDD(~pDi, ~pDf ) = 128i
2)−G5(pD1i + pD1f ) · (pD2i + pD2f )F
,(3.2)
where ps̄ and pD denote the four-momentum of the s̄-quark and diquark etc. The Fv
and Fs are the vector and scalar form factors of the scalar diquark. For simplicity,
we will assume that both take the dipole form, (1−q2/Λ2)−2, with Λ = 0.84 GeV. In
the NJL model calculation with the Pauli-Villars cutoff,8) the coupling constants are
related to the mesonic coupling constants by G1 = Gπ/2, G2 = Gρ/2 and G5 = Gω/2
which give Gv = −0.78 GeV
−2. We remark that the sign of Gv is definitely negative
since omega meson is heavier than the rho meson.
The potential matrix elements of Eqs. (3.1-3.2) can then be used in the scattering
equations obtained with the use of Blankenbecler-Sugar three-dimensional reduction
scheme7) for the BS equation for both the s̄D and DD systems. The resultant
scattering equations are solved to obtain the two-body t−matrix elements and the
phase shifts.
§4. Results for Θ+ and discussion
Our results for the phase shifts are shown in Fig. 3(a). We see that the s-wave
phase shifts for s̄D is positive which indicates the interaction is attractive, while the
p-wave DD interaction is repulsive since their phase shift is negative. In Fig. 3(b)
4 H. Mineo, J.A. Tjon, K. Tsushima, and S.N. Yang
we show the Gv dependence of the two-body s̄D binding energy. We see that with
the type of interaction constructed in Sec. 3, a s̄D bound state begins to appear only
when Gv becomes less than −5 ∼ −6 Gev
−2, far too negative as compared to the
physical value of -0.78 Gev−2 determined from the ρ− ω mass difference.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
pE [GeV](a)
6 8 10 12 14
-Gv [GeV](b)
Fig. 3. (a) Phase shifts δl for the s̄D in s-wave with Gv = −0.78 GeV
−2 (solid line) and DD in
p-wave (dashed line). (b) Gv dependence of the s̄D binding energy.
The three-body BSF equation7) takes the same form as the nonrelativistic one,
Ti(s) = ti(s) + ti(s)G0(s) [Tj(s) + Tk(s)] , (4.1)
where G0 is the free three-particle Green’s function and ti(s) is the two-particle
t−matrix of particles j and k with (i, j, k) being a cyclic permutation of (1, 2, 3). If
one uses the Blankenbecler-Sugar approximation for G0 and the two-body t−matrix
elements obtained in Sec. 3, then the homogeneous equation of Eq. (4.1) can be
solved9) to look for a possible three-body s̄DD bound state. We could not find
a bound pentaquark in JP = 1
channel. However, we do see a pentaquark in
JP = 1
channel when Gv becomes less that ∼ −8.0 GeV
−2. Pentaquark binding
energy EB(5q) grows from 77 to 505 MeV as Gv decreases from -8.0 to -14.0 GeV
The effect of consierdable weaker attraction in the JP = 1/2+ channel is caused by
the spectator particle being in a p-wave state.
Acknowledgements
One of the authors (S.N.Y.) thanks the Yukawa Institute for Theoretical Physics
at Kyoto University, for warm hospitality extended to him during the YKIS2006 on
”New Frontiers on QCD”.
References
1) T. Nakano, talk at YKIS2006 (Kyoto, Japan, November 20 - December 8, 2006).
2) T. Nakano et al. [LEPS Collaboration], Phys. Rev. Lett. 91, (2003) 012002.
3) R.L. Jaffe and F. Wilczek, Phys. Rev. Lett. 91, (2003) 232003.
4) Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1960); 124, 246 (1961).
5) S. Huang and J. Tjon, Phys. Rev. C49, 1702 (1994).
6) H. Mineo et al., Phys. Rev. C60, 065201 (1999); Nucl. Phys. A703, 785 (2002).
7) G. Rupp and J.A. Tjon, Phys. Rev. C37, (1988) 1729.
8) H. Mineo et al., Phys. Rev. C72, (2005) 025202.
9) A. Ahmadzadeh and J. Tjon, Phys. Rev. 147, (1966) 1111.
	Introduction
	SU(3)f NJL model and the diquark
	Two-body interactions for D and DD channels
	Results for + and discussion

  A Bethe-Salpeter-Faddeev (BSF) calculation is performed for the pentaquark
$\Theta^+$ in the diquark picture of Jaffe and Wilczek in which $\Theta^+$ is a
diquark-diquark-${\bar s}$ three-body system.
  Nambu-Jona-Lasinio (NJL) model is used to calculate the lowest order diagrams
in the two-body scatterings of ${\bar s}D$ and $D D$. With the use of coupling
constants determined from the meson sector, we find that ${\bar s}D$
interaction is attractive while $DD$ interaction is repulsive, and there is no
bound $\frac 12^+$ pentaquark state. A bound pentaquark $\Theta^+$ can only be
obtained with unphysically strong vector mesonic coupling constants.

Introduction
Experimental situation on the existence of Θ+ is currently filled with conflicting
positive and negative evidence and the question is still not yet fully settled.1)
Even if it turns out that Θ+ does not exist, it would still be theoretically interest-
ing to understand the absence of such an exotic. Many theoretical approaches have
been used, including quark models, QCD sum rules, and lattice QCD, in addition to
the chiral soliton model, to understand the properties and structure of Θ+ since its
first sighting.2) One of the most intriguing theoretical ideas suggested for Θ+ is the
diquark picture of Jaffe and Wilczek (JW)3) in which Θ+ is considered as a three-
body system consisted of two scalar, isoscalar, color 3̄ diquarks (D’s) and a strange
antiquark (s̄). It is based, in part, on group theoretical consideration. It would hence
be desirable to examine such a scheme from a more dynamical perspective.
It is known that diquark arises naturally from Nambu-Jona-Lasinio (NJL) model,
an effective quark theory in the low energy region.4) NJL model conveniently incor-
porates chiral symmetry and its spontaneously breaking which dictates the hadronic
physics at low energy. Models based on NJL type of Lagrangians have been very
successful in describing the low energy meson physics. Based on relativistic Fad-
∗) e-mail address: mineo@gate.sinica.edu.tw
∗∗) e-mail address: J.A.Tjon@phys.uu.nl
∗∗∗) e-mail address: tsushima@usal.es
†) e-mail address: snyang@phys.ntu.edu.tw
typeset using PTPTEX.cls 〈Ver.0.9〉
http://arxiv.org/abs/0704.1072v2
2 H. Mineo, J.A. Tjon, K. Tsushima, and S.N. Yang
deev equation, the NJL model has also been applied to the baryon systems.5), 6)
It has been shown that, using the quark-diquark approximation, one can explain
the nucleon static properties and the qualitative features of the empirical valence
quark distribution reasonably well.6) Consequently, we will employ NJL model to
describe the dynamics of a (s̄DD) three-particle system in Faddeev formalism. We
use relativistic equations to describe both the three-particle and its two-particle sub-
systems, namely, the Bethe-Salpeter-Faddeev (BSF) equation7) and Bethe-Salpeter
(BS) equations. In practice, Blankenbecler-Sugar reduction scheme is used to reduce
the four-dimensional integral equation into three-dimensional ones.
§2. SU(3)f NJL model and the diquark
The SU(3)f NJL model is a chirally symmetric four-fermi contact interaction
Lagrangian. With the use of Fierz transformations, the original NJL interaction
Lagrangian LI can be rewritten, for the qq̄ channel, as
LI,qq̄ = G1
(ψ̄λafψ)
2 − (ψ̄γ5λafψ)
(ψ̄γµλafψ)
2 + (ψ̄γµγ5λafψ)
(ψ̄γµλ0fψ)
2 + (ψ̄γµγ5λ0fψ)
(ψ̄γµλ0fψ)
2 − (ψ̄γµγ5λ0fψ)
+ · · · , (2.1)
where a = 0 ∼ 8, and λ0
I. For later use, we define G5 = G2 +
Gv , with
Gv ≡ G3 + G4. For the scalar, isoscalar diquark channel, interaction Lagrangian is
given by
LI,s = Gs
ψ̄(γ5C)λ2fβ
ψT (C−1γ5)λ2fβ
, (2.2)
where βAc =
λA(A = 2, 5, 7) corresponds to one of the color 3̄c states. C = iγ
is the charge conjugation operator, and λ′s are the Gell-Mann matrices.
The constituent quark and diquark masses can be obtained from the gap equation
and t-matrix of the diquark. Since we are only interested in a qualitatively study of
the interactions inside Θ+, we will use the empirical values of the constituent quark
masses Mu,d = 400 MeV, Ms = 600 MeV, and the diquark mass MD = 600 MeV as
obtained in Ref.8)
§3. Two-body interactions for s̄D and DD channels
In the JW model for Θ+,3) symmetry consideration requires that the the spatial
wave function of the two scalar-isoscalar, color 3̄ diquarks must be antisymmetric
and the lowest possible state is p-state. Since Θ+ is of JP = 1
, s̄ would be in
relative s-wave to the DD pair. Accordingly, we will consider only the configuration
where s̄D and DD are in relative s- and p-waves, respectively.
Fig. 1 shows the lowest order diagram, i.e., first order in LI,qq̄ in s̄D scat-
tering. Trace properties in Dirac and flavor space limit the vertex Γ to only the
vector-isoscalar term, (ψ̄γµλ0
ψ)2. For the DD interaction, the quark rearrangement
Faddeev Calculation for Pentaquark Θ+ 3
diagram gives no contribution because of its color structure. The lowest order non-
vanishing diagram of the first order in LI,qq̄ is given in Fig. 2 where only the direct
term is shown. The corresponding exchange diagram vanishes again because of the
color structure.
Γ=λ f
aγµ (a=0,8)
Fig. 1. s̄D potential of the lowest order
in LI,qq̄ .
Fig. 2. Lowest order diagrams in DD
scattering.
With the use of the interaction Lagrangians of Eqs. (2.1-2.2), we obtain the
following driving terms of Fig. 1 and 2, in the BS equations for s̄D and DD two-
particle systems,
< s̄fDf |V |s̄iDi > = (−v̄(ps̄i))(−iVs̄D)(pDi, pDf )v(ps̄f ),
Vs̄D =
GvFv(q
2)Ṽs̄D(pDi, pDf ), Ṽs̄D(pDi, pDf ) = (6pDi + 6pDf )/2 (3.1)
− iVDD(~pDi, ~pDf ) = 128i
2)−G5(pD1i + pD1f ) · (pD2i + pD2f )F
,(3.2)
where ps̄ and pD denote the four-momentum of the s̄-quark and diquark etc. The Fv
and Fs are the vector and scalar form factors of the scalar diquark. For simplicity,
we will assume that both take the dipole form, (1−q2/Λ2)−2, with Λ = 0.84 GeV. In
the NJL model calculation with the Pauli-Villars cutoff,8) the coupling constants are
related to the mesonic coupling constants by G1 = Gπ/2, G2 = Gρ/2 and G5 = Gω/2
which give Gv = −0.78 GeV
−2. We remark that the sign of Gv is definitely negative
since omega meson is heavier than the rho meson.
The potential matrix elements of Eqs. (3.1-3.2) can then be used in the scattering
equations obtained with the use of Blankenbecler-Sugar three-dimensional reduction
scheme7) for the BS equation for both the s̄D and DD systems. The resultant
scattering equations are solved to obtain the two-body t−matrix elements and the
phase shifts.
§4. Results for Θ+ and discussion
Our results for the phase shifts are shown in Fig. 3(a). We see that the s-wave
phase shifts for s̄D is positive which indicates the interaction is attractive, while the
p-wave DD interaction is repulsive since their phase shift is negative. In Fig. 3(b)
4 H. Mineo, J.A. Tjon, K. Tsushima, and S.N. Yang
we show the Gv dependence of the two-body s̄D binding energy. We see that with
the type of interaction constructed in Sec. 3, a s̄D bound state begins to appear only
when Gv becomes less than −5 ∼ −6 Gev
−2, far too negative as compared to the
physical value of -0.78 Gev−2 determined from the ρ− ω mass difference.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
pE [GeV](a)
6 8 10 12 14
-Gv [GeV](b)
Fig. 3. (a) Phase shifts δl for the s̄D in s-wave with Gv = −0.78 GeV
−2 (solid line) and DD in
p-wave (dashed line). (b) Gv dependence of the s̄D binding energy.
The three-body BSF equation7) takes the same form as the nonrelativistic one,
Ti(s) = ti(s) + ti(s)G0(s) [Tj(s) + Tk(s)] , (4.1)
where G0 is the free three-particle Green’s function and ti(s) is the two-particle
t−matrix of particles j and k with (i, j, k) being a cyclic permutation of (1, 2, 3). If
one uses the Blankenbecler-Sugar approximation for G0 and the two-body t−matrix
elements obtained in Sec. 3, then the homogeneous equation of Eq. (4.1) can be
solved9) to look for a possible three-body s̄DD bound state. We could not find
a bound pentaquark in JP = 1
channel. However, we do see a pentaquark in
JP = 1
channel when Gv becomes less that ∼ −8.0 GeV
−2. Pentaquark binding
energy EB(5q) grows from 77 to 505 MeV as Gv decreases from -8.0 to -14.0 GeV
The effect of consierdable weaker attraction in the JP = 1/2+ channel is caused by
the spectator particle being in a p-wave state.
Acknowledgements
One of the authors (S.N.Y.) thanks the Yukawa Institute for Theoretical Physics
at Kyoto University, for warm hospitality extended to him during the YKIS2006 on
”New Frontiers on QCD”.
References
1) T. Nakano, talk at YKIS2006 (Kyoto, Japan, November 20 - December 8, 2006).
2) T. Nakano et al. [LEPS Collaboration], Phys. Rev. Lett. 91, (2003) 012002.
3) R.L. Jaffe and F. Wilczek, Phys. Rev. Lett. 91, (2003) 232003.
4) Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1960); 124, 246 (1961).
5) S. Huang and J. Tjon, Phys. Rev. C49, 1702 (1994).
6) H. Mineo et al., Phys. Rev. C60, 065201 (1999); Nucl. Phys. A703, 785 (2002).
7) G. Rupp and J.A. Tjon, Phys. Rev. C37, (1988) 1729.
8) H. Mineo et al., Phys. Rev. C72, (2005) 025202.
9) A. Ahmadzadeh and J. Tjon, Phys. Rev. 147, (1966) 1111.
	Introduction
	SU(3)f NJL model and the diquark
	Two-body interactions for D and DD channels
	Results for + and discussion
