Triquark structure and isospin symmetry breaking in
exotic Ds mesons
S. Yasui and M. Oka
Department of Physics, Tokyo Institute of Technology,
Tokyo 152-8551, Japan
January 14, 2019
Abstract
The color anti-triplet triquark qq̄q̄ is considered as a compact component in the
tetraquark structure cqq̄q̄ of exotic Ds mesons. We discuss the mass spectrum and
the flavor mixing of the triquarks by using the instanton induced interaction and
the one-gluon exchange potentials. As a characteristic property of the triquark, we
investigate the isospin violation. It is shown that the flavor 3 (isosinglet) and 6
(isotriplet) states may be strongly mixed and then are identified with Ds(2632).
1 Introduction
Exotic Ds mesons attract much attention recently. The BaBar Collaboration [1] first
announced the Ds(2317) with J
π = 0+, whose mass lies approximately 160 MeV below
the constituent quark model predictions [2, 3]. The decay width is less than 4.6 MeV.
This state was confirmed by CLEO [4] and Belle [5]. It was followed by Ds(2460) with
1+, reported by CLEO [4], and Ds(2632) by SELEX [6]. At the same time, a charmonium
candidate X(3872) was reported by Belle [7] and the other groups. These new mesons are
novel because they do not fit to the quark model expectations and have caused further
studies and speculations on their structures. In particular, it has been proposed that some
of them are excited two-quark states [8, 9], chiral doublets in heavy quark limit [10], the
molecular bound states [11, 12] or tetraquarks cqq̄q̄ (q = u, d and s) [13, 14, 15, 16, 17,
18, 19].
Here we briefly review the previous researches about the molecular and tetraquark
pictures. Some suggest that the Ds(2317) may be a DK molecule state, and the Ds(2460)
a D∗K molecule. Indeed the masses of the Ds(2317) and Ds(2460) are slightly below the
thresholds of theD+K andD∗+K, respectively. The mass splitting 140 MeV between Ds
and D∗s is almost the same as that between the Ds(2317) and Ds(2460). The ground state
of Ds is the Ds(1969) with 0
− and the Ds(2112) with 1
−. Therefore, the Ds(2317) can
be assigned to be the ground state in the 0+ sector, while the Ds(2460) can be identified
http://arxiv.org/abs/0704.1345v3
also as a ground state in the 1+ sector. These properties are also explained in the chiral
effective theory [10].
One of the prominent properties of the exotic Ds mesons is the isospin violating decay
process Ds(2317) → Dsπ0 [1]. This process is considered to be realized by the virtual η
emission and the η − π mixing, since the Ds(2317) is supposedly an isosinglet state.1 An
anomalous branching ratio Γ(Ds(2632) → D0K+)/Γ(Ds(2632) → Dsη) ≃ 0.16 ± 0.06 [6]
is also a very interesting problem. In the conventional cs̄ picture, the decay to the D0K+
is favored as compared with the decay to Dsη, since uū or dd̄ creation would be easier
than ss̄. Maiani et al. [17] considered the tetraquark state cs̄dd̄, in which the isospin is
maximally violated. This state has decay modes of [cs̄][dd̄] (Dsη) and [cd̄][ds̄] (D
+K0),
while the D0K+ decay is suppressed by the OZI forbidden process dd̄→ uū. This picture
was also applied to study of the isospin violation in the decay process of X(3872) [18].
On the other hand, Chen and Li considered cs̄ss̄ [14]. They discussed that the decay to
Dsη is dominant, while the decay to D
0K+ is suppressed by 1/Nc due to the OZI rule and
the reduction of the amplitude due to the color matching to create a color singlet state.
Liu and Zhu discussed that the Ds(2632) is assigned as an isosinglet member in flavor 15
[19].
Let us see the possibility of the isospin violated eigenstates [17, 18]. When the quarks
have sufficiently large momenta, the asymptotic freedom suppresses the qq̄ creation pro-
cess, and the flavor mixing interaction is less important. Then, the alignment in the
diagonal components in the mass matrix realizes uū and dd̄ separately as eigenstates,
which are mixed states of isosinglet and isotriplet states. In general, however, the flavor
mixing term is in the order of ∼100 MeV [20, 21, 22, 23, 24], and much larger than the
mass difference between u and d quarks, |mu −md| <∼ 5 MeV. Therefore, it is expected
that the isospin breaking effect is too small to separate uū and dd̄.
The purpose of this paper is to discuss the microscopic mechanism of the isospin viola-
tion of the tetraquark for the open charm system, cqq̄q̄. It is noticed that the interaction
between the light quarks q and the c quark is suppressed in the heavy quark limit. Thus,
it is natural to consider that, in the first approximation, three light quarks are decoupled
from the heavy quark. Therefore we consider states compound by the u, d and s quarks
as triquarks or color non-singlet baryons. Here it must be noticed such a state cannot
exist as an asymptotic state, but only in the bound state.
We here consider a simple model with non-relativistic valence quarks under the influ-
ence of the one-gluon exchange (OGE) and the instanton induced interaction (III). The
mass spectroscopy of the triquark was first discussed in the diquark-triquark picture in the
literature of the pentaquark [25, 26, 27], and further investigated in details in the OGE in-
teraction [28, 29, 30]. Furthermore, the ’t Hooft interaction induced by the instanton was
also used [31, 32, 33]. However, the effective interaction employed in [31, 32, 33] operates
only in spin singlet and isosinglet channel in qq̄ pair, while the effective interaction used
in [20, 21, 22, 23, 24] operates, not only in spin singlet and isosinglet channel, but also in
spin triplet and isotriplet channel. It is known that the difference causes a discrepancy in
1Hayashigaki and Terasaki considers a possibility of the isotriplet state for Ds(2317) [16].
the meson mass spectrum [34]. Therefore it is an interesting problem to investigate the
isospin violation of the triquark by using the effective interaction in [20, 21, 22, 23, 24].
The content of this paper is as follows. In Section 2, the flavor representation of the
triquark, the III and OGE potential and the mass matrix are discussed. In Section 3, the
isospin mixing is investigated by considering the ud quark mass difference. In Section 4,
our discussion is summarized.
2 Quark model
In the tetraquark picture of the exotic Ds mesons, the triquark is considered as a bound
state composed by three light flavor quarks. The hamiltonian of the triquark is obtained
only in light flavors space, since the interaction between the light and heavy quarks
is suppressed in the OGE potential. This is also the case for the instanton induced
interaction, since the heavy quark has no zero mode and free from the instanton vacuum
[36, 37]. The flavor SU(3) multiplets of the triquark state qq̄q̄ are given as
3⊗ 3⊗ 3 = 3S ⊕ 3A ⊕ 6A ⊕ 15S.
We write the subscripts of S and A according to the symmetry under the exchange of two
anti-quarks. In Fig. 1, we show the weight diagrams of these multiplets. It is assumed
that all the quarks and anti-quarks occupy the lowest energy single particle orbital, the
s-wave orbital. In the following, we omit the subscripts in 6A and 15S for simplicity.
The exotic states reported in experiments have the strangeness S = +1. Then, the
isospin for each flavor multiplet is as follows; isosinglet for 3A, 3S and 15
, and isotriplet
for 6 and 15
. Here the isospin components of 15 are distinguished by the superscript. It
is straightforward to write down the flavor wavefunctions of these multiplets for S = +1.
isosinglet


|3A〉 = 12
u(s̄ū− ūs̄)− d(d̄s̄− s̄d̄)
|3S〉 = 12√2
2ss̄s̄+ u(s̄ū+ ūs̄) + d(d̄s̄+ s̄d̄)
|150〉 = 1
2ss̄s̄− u(s̄ū+ ūs̄)− d(d̄s̄+ s̄d̄)
isotriplet
|6〉 = 1
u(s̄ū− ūs̄) + d(d̄s̄− s̄d̄)
|151〉 = 1
u(s̄ū+ ūs̄)− d(d̄s̄+ s̄d̄)
In the following discussion, we consider only the S = +1 sector.
The triquark must belong to the color anti-triplet state, 3
S and 3
A, so that the
tetraquark is a color singlet state. Then, the spin and color combination of the tri-
quark is restricted by the Pauli principle. For example, the spin and color basis for the
flavor 3A and 6 states with the spin J = 1/2 is {|λX〉, |ρY 〉}. Here, λ and ρ stands for
the mixed states with λ- and ρ-symmetry in spin 1/2, and X and Y for color 3
S and 3
S S S
3 6 15
Figure 1: The weight diagram of the flavor SU(3) multiplets of the triquark qq̄q̄.
respectively. On the other hand, the basis for the flavor 3S and 15 states with spin 1/2 is
{|ρX〉, |λY 〉}. For spin J=3/2 state, we have |J=3/2 X〉 for 3A and 6, and |J =3/2 Y 〉
for 3S and 15.
Now we discuss the hamiltonian of the triquark. The instanton induced interaction
(III) has played very important role in the QCD vacuum in accompany with dynamical
chiral symmetry breaking [20, 21, 22, 23, 24]. It induces the Kobayashi-Kondo-Maskawa-
’t Hooft (KKMT) interaction [35, 36, 37], which is given as 2Nf point-like vertex with
the flavor anti-symmetric channel. In the quark model, the instanton effect has been
discussed in the non-relativistic limit in the KKMT interaction. The OGE potential is
also often used as an effective interaction [38]. Here we consider a hybrid model of the III
and OGE potentials [20, 21, 22, 23, 24]. The hamiltonian is
H = K + pIII
III +H
+ (1− pIII)VOGE +Mmass + Vconf , (2)
with the kinetic term K, the instanton induced interaction H
III (i = 2 and 3 for the two-
and three-body interactions), the OGE potential VOGE, the mass matrix Mmass and the
confinement potential Vconf . The parameter pIII controls the ratio of the III and the OGE
potentials. In the present discussion, we are interested in the isospin symmetry breaking,
and not involved with the absolute masses of the tetraquarks. Therefore, we pick up only
the III and OGE terms and the mass matrix;
H̃ = pIII
III +H
+ (1− pIII)VOGE +Mmass. (3)
Since we do not solve the quark confinement dynamically, we just use a quark wave
function from the harmonic oscillator potential with frequency ω.
Concerning the III potential, the three body force in the three quark state, q1q2q3, is
given in the flavor diagonal form as
III =
~λ1 ·~λ2 + ~λ2 ·~λ3 + ~λ3 ·~λ1
dabcλ
~σ1 ·~σ2~λ1 ·~λ2 + ~σ2 ·~σ3~λ2 ·~λ3 + ~σ3 ·~σ1~λ3 ·~λ1
0 −〈ψ̄ψ〉 mq ω αs
-0.2564 [GeV fm3] (0.25)3 [GeV3] 0.3837 [GeV] 0.5 [GeV] 1.319
Table 1: The parameter set from [21].
dabcλ
3 (~σ1 ·~σ2 + ~σ2 ·~σ3 + ~σ3 ·~σ1)
ǫijkσ
3fabcλ
δ(3)(~r1 − ~r2)δ(3)(~r2 − ~r3),
with a coupling constant V
0 and the delta functions as a point-like three body interaction.
We can deduce the two body instanton induced force,
III =
~λi ·~λj +
~σi ·~σj~λi ·~λj
δ(3)(~ri − ~rj),
using the quark condensate 〈ψ̄ψ〉, where the coupling constant V (2)0 is given as
〈ψ̄ψ〉V (3)0 .
The interactions in the q1q̄2q̄3 state are also obtained in a straightforward way.
The OGE potential between the q1q2 pair is given as
VOGE = 4παS
~λ1 ·~λ2
− ~σ1 ·~σ2
6m1m2
with a coupling constant αS. The first term is the electric interaction, and the second the
magnetic interaction with spin dependence. However, we neglect the electric interaction,
since in general it is sufficiently small as compared with the magnetic interaction. It should
be noted that the magnetic interaction is switched off with a suppression of 1/mQ for the
heavy-light quark pair (Qq) in the limit of the heavy mass. Therefore, it is understood
that the triquark qq̄q̄ may exist as a compound unit in the tetraquark structure. As a
summary, our interaction is sketched in Fig. 2. The parameter set in our interaction [21]
is summarized in Table 1.
From the III and OGE potentials, the energy spectrum of the triquark is obtained
in the following way. By using the basis of the spin and color, {|λX〉, |ρY 〉, |ρX〉, |λY 〉},
we obtain the hamiltonian in matrix forms for flavor 3A, 3S, 6 and 15 representations,
respectively.
First we consider the III potential. For the flavor 3A and 3S states, the hamiltonian
is given in the basis {|λX〉, |ρY 〉, |ρX〉, |λY 〉} by
HIII(3) =

0 3 −3

0 I2 +

0 0 0 0
0 0 0 0

0 I3, (4)
(c) qq 
(d) qq 
(e) qq (annihilation) 
(a) two-body (qq) 
(b) three-body (qqq) 
Figure 2: The diagram contributions for the III and OGE potentials. III: (a) the two-body
interaction for qq̄ and (b) the three body interaction for qq̄q̄. OGE: (c) qq̄, (d) q̄q̄ and (e) qq̄
(annihilation).
where I2 and I3 are the expectation values of the delta function for the point-like inter-
action,
I2 = 〈Ψ|δ(3)(r1 − r2)|Ψ〉 =
for the two-body interaction, and
I3 = 〈Ψ|δ(3)(r1 − r2)δ(3)(r2 − r3)|Ψ〉 =
for the three-body interaction with the triquark spatial wavefunction Ψ. It should be
mentioned that the 3A and 3S states are mixed due to the off-diagonal element in the two-
body interaction in the III potential, since {|λX〉, |ρY 〉} belongs to 3A and {|ρX〉, |λY 〉}
to 3S. Therefore, we may denote the mixed state as 3 in the following discussion. In the
similar way, for the flavor 6 state, the basis {|λX〉, |ρY 〉} gives the matrix
HIII(6) =
0 I2 +
0 I3, (7)
and for the flavor 15 state, the basis {|ρX〉, |λY 〉} gives
HIII(15) =
0 I2. (8)
Second we consider the OGE potential. For the 3 state, we obtain the matrix in the
basis of {|λX〉, |ρY 〉, |ρX〉, |λY 〉},
VOGE(3) =


I2. (9)
Furthermore, for the 6 state, we obtain in the basis {|λX〉, |ρY 〉}
VOGE(6) =
I2, (10)
and for the 15 state
VOGE(15) =
I2, (11)
in the basis {|ρX〉, |λY 〉}.
The mass differences among u, d and s quarks induce mixings between the flavor
representations. In the basis of the flavor representation, {|3A〉, |3S〉, |15
0〉, |6〉, |151〉}, we
easily obtain the mass part of the hamiltonian for S = +1 sector, as
Mmass =

mu +md +ms 0 0 mu −md 0
0 mu+md
+ 2ms −mu+md2 +ms 0
mu−md√
0 −mu+md
+ms 0 0 −mu−md√2
mu −md 0 0 mu +md +ms 0
0 mu−md√
−mu−md√

.(12)
The diagonal elements are isosinglet and isotriplet components, while the off-diagonal
elements induce mixings between them. Note that the flavor representations with the
same symmetry (A or S) are mixed. The 3A and 15
states are mixed with each other
by the SU(3) symmetry breaking ( mu = md < ms). We also note that the isosinglet
states (3A, 3S, 15
) and the isotriplet states (6, 15
) are also mixed due to the isospin
symmetry breaking (mu < md)
2. We consider this interaction as a driving force for the
isospin symmetry breaking in the next section. It should be noted that the Coulomb or
electromagnetic interaction may also break isospin symmetry, which is not considered in
this study.
3 Isospin mixing
In general, the u − d quark mass difference is sufficiently small as compared with the
energy splitting between the isosinglet and isotriplet states, and the isospin breaking can
2In the works in [31, 32, 33], the 3A and 6 states are mixed due to the mu = md 6= ms. As long as
the isospin symmetry is not violated, however, we have no mixing between the 3A and 6 states.
be neglected. However, in the triquark, we see that the isosinglet and isotriplet states
sometimes happen to be degenerate and thus a large isospin mixing can occur. In this
section, we investigate the mixing of the isosinglet and isotriplet states. For this purpose,
we calculate the eigenenergies, E, of the hamiltonian (3). We choose the s quark mass
ms = 0.48 GeV and the strength of the harmonic potential ω = 0.50 GeV in the following
discussion. We take the parameter pIII as a free parameter.
We present the binding energy spectrum of the triquark, ∆E = E − (mu +md +ms),
for the OGE (pIII = 0) and III (pIII = 1) potentials in Fig. 3. The isosinglet and isotriplet
states are shown by the solid and dashed lines, respectively. As the J = 3/2 states are
heavier than J = 1/2, in the following discussion, we pay attention to the ground states
with the spin J=1/2, the 3 and 6 multiplets.
Let us see the result by the III potential. In SU(3) symmetric case, the ground state
is the 3 state, which contains mainly the 3A component rather than the 3S component.
On the other hand, the 3S component is mixed in the excited state in the 3 state. Now
we break the SU(3) symmetry with keeping the isospin symmetry; mu = md < ms. The
ground state is still the 3 state, and the first excited state is the 15
state, followed by
the 15
and 6 states. The splitting between the 15
and 15
states makes the former
lifted up as compared with the latter. This splitting comes from the fact that the mass
matrix (12) mixes the 3S and 15
states. The same mixing pushes the 3 upward.
On the other hand, in the OGE potential, the ground state is the 6 state, followed by
the 3, 15
, and 15
states. It should be noticed that the flavor multiplets are different
in the III and the OGE potentials. Especially the change of the ground state flavor is
important for the isospin symmetry breaking as we see below.
The reason that the 6 state is the ground state in the OGE can be understood by
examining the annihilation diagram in Fig. 2(e). It vanishes for the usual color singlet
meson qq̄, since the gluon (g) contained in the process qq̄ → g → qq̄ is a color octet state.
In the triquark, however, the annihilation diagram does not vanish. This is because the
gq̄ state contained in the process qq̄q̄ → gq̄ → qq̄q̄ remains color anti-triplet 3c due to the
color decomposition,
c ⊗ 3c = 3c ⊕ 6c ⊕ 15c.
Note that the initial and final qq̄q̄ states are also color anti-triplet. The annihilation term
increases the energy of the flavor 3 state, while it does not operate for the 6 state (see Eq.
(1) ). Consequently, the 3 state is about 50 MeV above the 6 state in the OGE potential.
Let us return to the discussion of the isospin symmetry breaking. We recall that the
3 state is isosinglet and the 6 state is isotriplet. Thus the mixing of the flavor multiplets
are directly related to the isospin mixing. Explicitly, we plot the binding energies of the
flavor multiplets as functions of the parameter pIII in Fig. 4. The solid lines indicate the
isosinglet states, and the dashed lines the isotriplet states. We find that the isosinglet
and isotriplet states become degenerate at A (pIII = 0.18) and B (pIII = 0.82).
Now let us introduce the isospin symmetry breaking, namely the ud quark mass differ-
ence, ∆m = md −mu ≃ 0.005 GeV [39]. The ∆m is comparable to the energy difference
between the isosinglet and isotriplet states at A and B. There, the two degenerate states
ΔE [GeV]
(p  =0)
(p  =1)
SU(3)
 breaking
SU(3)
 symmetry
SU(3)
 breaking
SU(3)
 symmetry
III III
isosinglet
isotriplet
Figure 3: The binding energies of the triquarks with J = 1/2 for the OGE (pIII = 0) and III
(pIII = 1) potentials.
0 0.5 1
I=0 (3bar, 15bar)
I=1 (6)
I=1 (15bar)
Figure 4: The binding energies of the various flavor multiplets with SU(3) breaking as functions
of the parameter pIII . The bold-solid line indicates the isosinglet (3 and 15
) states, the bold-
dashed line isotriplet (6), and the thin-dashed line the isotriplet (15
) states. Cf. Fig. 3.
0 0.1 0.2 0.3 0.4
uubar
ddbar
ssbar
0.6 0.7 0.8 0.9 1
uubar
ddbar
ssbar
Figure 5: The ratios of the isosinglet (bold-solid line) and isotriplet (bold-dashed line) states
as functions of the parameter pIII . (a) and (b) corresponds to the state A and B, respectively,
in Fig. 4. The ratios of the uū (thin-solid line), dd̄ (thin-dashed line) and ss̄ (thin-dot-dashed
line) are also shown.
will split into two isospin mixed states which are orthogonal to each other. Here we choose
one state at A. The ratios of isosinglet and isotriplet components are plotted as functions
of the parameter pIII in Fig. 5(a). In the range of 0.16 < pIII < 0.20, we see a rapid
change of the isosinglet (bold-solid line) and the isotriplet (bold-dashed line) components,
hence the isospin is strongly mixed. In the same way at B, we also see an isospin mixing
at pIII = 0.82 as shown in Fig. 5(b). However, in contrast to the case A, the isospin
mixing at B occurs in a small range of the parameter pIII . This is understood from the
mass matrix (12). At A, the isosinglet state is almost the 3A multiplet, while the isotriplet
state is purely the 6 multiplet (see Fig.3). The mass matrix (12) induces the 3A and 6
multiplet mixing, namely the isospin violation, by mu−md. On the other hand, at B, the
isosinglet state is changed to be the 15
state, while the isotriplet state is the same. In the
mass matrix (12), however, there is no direct mixing between the 15
and 6 multiplets.
They are mixed indirectly through the multi-step mixings of the 6 − 3A, 3A − 3S, and
3S − 15
. Therefore the isospin mixing at B is suppressed as compared to that at A.
Here we recall the isospin violation in experimental observations. Maiani et al. con-
sider Ds(2632) as the cs̄dd̄ state, which is an isospin mixed state [17]. In our analysis, the
isospin mixing at A induces a mixing between the isosinglet (mostly 3A) and isotriplet (6)
states. Hence, from Eq. (1), the mixed wavefunction, |3A〉 − |6〉 = −d(d̄s̄− s̄d̄), contains
only the dd̄ component. On the other hand, at B, there is a mixing between the isosinglet
(mostly 15
) and the isotriplet (6) states. There, from Eq. (1), the mixed wavefunction
|150〉 − |6〉 contains both of the uū and dd̄ components with the same fraction. Conse-
quently, we see that the isospin mixed states, cs̄uū and cs̄dd̄, become separate eigenstates
by the 3− 6 mixing rather than the 150 − 6 mixing.
We also understand this result explicitly by looking at the fraction of uū, dd̄ and ss̄
components at A and B in Fig. 5(a) and (b), respectively. In Fig. 5(a), the dd̄ fraction
(thin-dashed line) is overwhelming as compared with the uū fraction (thin-solid line)
around pIII = 0.18. In contrast, in Fig. 5(b), the fraction of the uū and dd̄ components
are almost the same at pIII = 0.82. Therefore, the isospin mixed state at A gives the
cs̄dd̄, while the state at B does not. Thus, the discussion by Maiani et al. in [17] is proven
to be possible as the the 3− 6 mixing.
So far, we have discussed the isospin mixing by using the isospin basis of isosinglet
and isotriplet. However, the isospin mixing is also investigated by basis {uū, dd̄}. Then
the hamiltonian is generally given by
(uū dd̄
uū m δ
dd̄ δ m+ 2∆m
. (13)
The uū and dd̄ are eigenstates of this hamiltonian, if the flavor mixing term δ is much
smaller than ∆m, and only the diagonal component is dominant. However, in general, δ
is in the order of hundred MeV in the vacuum as we see the mass splitting of π − η.
For the triquark with J = 1/2, due to the combination of spin and color, we have four
uū-like states and also four dd̄-like states. In this basis, the hamiltonian is given by
uū dd̄

m1 0 δ11 δ12 δ13 δ14
m2 δ21 δ22 δ23 δ24
m3 δ31 δ32 δ33 δ34
0 m4 δ41 δ42 δ43 δ44
δ11 δ12 δ13 δ14 m1+2∆m 0
δ21 δ22 δ23 δ24 m2+2∆m
δ31 δ32 δ33 δ34 m3+2∆m
δ41 δ42 δ43 δ44 0 m4+2∆m

where the diagonal uū−uū and dd̄−dd̄ parts are diagonalized in the spin and color spaces.
The diagonalized energy, m1, m2, m3 and m4, are plotted as functions of the parameter
pIII in Fig. 6(a). If the flavor mixing strength δij (i, j = 1, · · · , 4) are sufficiently small,
the lowest uū- and dd̄-like states become eigenstates. As shown in Fig. 6(b), δ11 is so
small as compared with ∆m around pIII = 0.18. There, the eigenstate become uū- and
dd̄-like states, hence the isosinglet and isotriplets states are ideally mixed. This result is
consistent with our discussion that the isospin mixing is caused by the 3−6mixing around
pIII = 0.18. It should be noted that the contribution from the higher states is suppressed
since the mixing is in the order of δij/(mk−m1) ≃ 0.1 (k ≥ 2) in the perturbation theory.
Therefore the first order perturbation is sufficient for the present discussion.
Lastly, we discuss the parameter dependence of the isospin mixing. We employ several
free parameters, the s quark mass ms, the harmonic oscillator potential frequency ω and
the parameter pIII for the OGE and III potentials. They may have some uncertainty due
to the lack of the experimental information. However, one sees that the results are not
0 0.5 1
0 0.1 0.2 0.3 0.4
Figure 6: (a) The diagonal components of the uū − dd̄ matrix as functions of the parameter
pIII . (b) δ11 as a function of pIII . Note the energy unit is given by MeV in (b). See the text.
modified qualitatively by parameter change. As an example, we plot the size parameter
b = 1/
mqω of the triquark wavefunction, which causes the 3 − 6 mixing at A, as a
function of the parameter pIII for ms = 0.48 GeV and 0.58 GeV. We see that the b comes
within a reasonable range 0.4 < b < 0.6 fm, and is not far from b = 0.5 fm [21]. This
range is little affected by ms.
Concerning the range of the parameter pIII , the obtained value pIII = 0.18 is smaller
than the conventionally used value pIII ≃ 0.4 in hadron spectroscopy [21]. This observa-
tion indicates that the OGE is more dominant than the instanton induced interaction in
tetraquarks. It is noticed that the value pIII = 0.18 is not obtained dynamically, since the
quark wave function is assumed to be Gaussian. The present study suggests that there
would exist an essential mechanism to choose such pIII in charmed tetraquark.
When the linear potential is used as a confinement potential, the quark wave function
is modified from that of the harmonic oscillator potential, and the absolute values of
the OGE and the instanton induced interaction are also modified. However, the ratio of
both interactions is not changed, since both of the potentials are point-like interactions.
In the present discussion, the isospin symmetry breaking is induced by the ratio of two
interactions. Therefore our conclusion is not modified qualitatively.
4 Conclusion
Possibility of isospin violation in the Ds tetraquark systems is examined in this paper.
Tetraquarks are candidates of the exotic Ds mesons recently reported in experiment. We
consider the energy spectrum of the triquark by using the non-relativistic quark model
with the instanton induced interaction and the one-gluon exchange potentials. With
taking the SU(3) symmetry breaking into account for S = +1 sector, we show that the
flavor 3 (isosinglet) and the flavor 6 (isotriplet) representations form the ground states.
Considering the isospin symmetry breaking by the quark mass difference, mu < md, it is
0 0.5 1
ms=0.48 [GeV]
ms=0.58 [GeV]
Figure 7: The b− pIII relation for the 3− 6 mixing. ms = 0.48 GeV (solid line) and 0.58 GeV
(dashed line). See the text.
shown that the 3 (isosinglet) and 6 (isotriplet) states may be mixed strongly with some
range of the parameter pIII . There the isosinglet and the isotriplet states are ideally
mixed, and one of the eigenstates is dominated by the dd̄ component. This result is
also investigated by looking at the off-diagonal components in the uū − dd̄ matrix. Our
conclusion supports the discussion given in [17, 18].
How do we experimentally confirm the picture given in this paper? The present
mechanism of the isospin symmetry violation relies on the suppression of the flavor mixing
interaction. Thus, at the ideal (maximal) mixing, the uū- and dd̄-like states are split by
the diagonal part of the mass matrix, namely by 2∆m ∼ 10 MeV. Therefore the two states
are expected to come close to each other. So far, due to the experimental restriction, only
a few charged decay modes are observed, and they suggest a dd̄-like state, D+s (2630/cs̄dd̄),
where its main decay mode is D+s (2630) → Dsη, while D+s (2630) → D0K+ is suppressed.
The corresponding uū-like state will show different decay patterns. Therefore careful
analyses of different charged modes of decays will reveal the nature of the isospin breaking.
In particular, the decays into Dsπ
0 and D+K0 are two interesting modes.
The present study suggests the possibility of the triquark à la “color non-singlet
baryon”, which is a color non-singlet particle composed by three quarks. Although the
triquark itself cannot exist asymptotically, it may appear as an effective degree of freedom
in the exotic heavy mesons in heavy quark mass limit. It is considered in general that
the color non-singlet light quark systems may exist by color neutralization with heavy
quark spectator [30]. The triquark is a possible candidate among the color non-singlet
quark systems, which can be examined by studying the tetraquark structure of exotic
open charm mesons. The triquark would be also an interesting object in the lattice QCD
simulation. Furthermore the triquark may be a relevant degree of freedom as a color non-
singlet compound particle in the deconfinement phase such as the quark-gluon plasma and
the quark matter. In order to understand such states in many aspects, it is important to
study several properties, such as masses, decay widths and so forth.
Acknowledgment
We express our thanks to Dr. T. Shinozaki and Prof. S. Takeuchi for discussions. This
work is supported by a Grant-in-Aid for Scientific Research for Priority Areas, MEXT
(Ministry of Education, Culture, Sports, Science and Technology) with No. 17070002.
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	Introduction
	Quark model
	Isospin mixing
	Conclusion

  The color anti-triplet triquark $q\bar{q}\bar{q}$ is considered as a compact
component in the tetraquark structure $cq\bar{q}\bar{q}$ of exotic $D_{s}$
mesons. We discuss the mass spectrum and the flavor mixing of the triquarks by
using the instanton induced interaction and the one-gluon exchange potentials.
As a characteristic property of the triquark, we investigate the isospin
violation. It is shown that the flavor $\bar{\bf 3}$ (isosinglet) and $\bf 6$
(isotriplet) states may be strongly mixed and then are identified with
$D_{s}(2632)$.

Introduction
Exotic Ds mesons attract much attention recently. The BaBar Collaboration [1] first
announced the Ds(2317) with J
π = 0+, whose mass lies approximately 160 MeV below
the constituent quark model predictions [2, 3]. The decay width is less than 4.6 MeV.
This state was confirmed by CLEO [4] and Belle [5]. It was followed by Ds(2460) with
1+, reported by CLEO [4], and Ds(2632) by SELEX [6]. At the same time, a charmonium
candidate X(3872) was reported by Belle [7] and the other groups. These new mesons are
novel because they do not fit to the quark model expectations and have caused further
studies and speculations on their structures. In particular, it has been proposed that some
of them are excited two-quark states [8, 9], chiral doublets in heavy quark limit [10], the
molecular bound states [11, 12] or tetraquarks cqq̄q̄ (q = u, d and s) [13, 14, 15, 16, 17,
18, 19].
Here we briefly review the previous researches about the molecular and tetraquark
pictures. Some suggest that the Ds(2317) may be a DK molecule state, and the Ds(2460)
a D∗K molecule. Indeed the masses of the Ds(2317) and Ds(2460) are slightly below the
thresholds of theD+K andD∗+K, respectively. The mass splitting 140 MeV between Ds
and D∗s is almost the same as that between the Ds(2317) and Ds(2460). The ground state
of Ds is the Ds(1969) with 0
− and the Ds(2112) with 1
−. Therefore, the Ds(2317) can
be assigned to be the ground state in the 0+ sector, while the Ds(2460) can be identified
http://arxiv.org/abs/0704.1345v3
also as a ground state in the 1+ sector. These properties are also explained in the chiral
effective theory [10].
One of the prominent properties of the exotic Ds mesons is the isospin violating decay
process Ds(2317) → Dsπ0 [1]. This process is considered to be realized by the virtual η
emission and the η − π mixing, since the Ds(2317) is supposedly an isosinglet state.1 An
anomalous branching ratio Γ(Ds(2632) → D0K+)/Γ(Ds(2632) → Dsη) ≃ 0.16 ± 0.06 [6]
is also a very interesting problem. In the conventional cs̄ picture, the decay to the D0K+
is favored as compared with the decay to Dsη, since uū or dd̄ creation would be easier
than ss̄. Maiani et al. [17] considered the tetraquark state cs̄dd̄, in which the isospin is
maximally violated. This state has decay modes of [cs̄][dd̄] (Dsη) and [cd̄][ds̄] (D
+K0),
while the D0K+ decay is suppressed by the OZI forbidden process dd̄→ uū. This picture
was also applied to study of the isospin violation in the decay process of X(3872) [18].
On the other hand, Chen and Li considered cs̄ss̄ [14]. They discussed that the decay to
Dsη is dominant, while the decay to D
0K+ is suppressed by 1/Nc due to the OZI rule and
the reduction of the amplitude due to the color matching to create a color singlet state.
Liu and Zhu discussed that the Ds(2632) is assigned as an isosinglet member in flavor 15
[19].
Let us see the possibility of the isospin violated eigenstates [17, 18]. When the quarks
have sufficiently large momenta, the asymptotic freedom suppresses the qq̄ creation pro-
cess, and the flavor mixing interaction is less important. Then, the alignment in the
diagonal components in the mass matrix realizes uū and dd̄ separately as eigenstates,
which are mixed states of isosinglet and isotriplet states. In general, however, the flavor
mixing term is in the order of ∼100 MeV [20, 21, 22, 23, 24], and much larger than the
mass difference between u and d quarks, |mu −md| <∼ 5 MeV. Therefore, it is expected
that the isospin breaking effect is too small to separate uū and dd̄.
The purpose of this paper is to discuss the microscopic mechanism of the isospin viola-
tion of the tetraquark for the open charm system, cqq̄q̄. It is noticed that the interaction
between the light quarks q and the c quark is suppressed in the heavy quark limit. Thus,
it is natural to consider that, in the first approximation, three light quarks are decoupled
from the heavy quark. Therefore we consider states compound by the u, d and s quarks
as triquarks or color non-singlet baryons. Here it must be noticed such a state cannot
exist as an asymptotic state, but only in the bound state.
We here consider a simple model with non-relativistic valence quarks under the influ-
ence of the one-gluon exchange (OGE) and the instanton induced interaction (III). The
mass spectroscopy of the triquark was first discussed in the diquark-triquark picture in the
literature of the pentaquark [25, 26, 27], and further investigated in details in the OGE in-
teraction [28, 29, 30]. Furthermore, the ’t Hooft interaction induced by the instanton was
also used [31, 32, 33]. However, the effective interaction employed in [31, 32, 33] operates
only in spin singlet and isosinglet channel in qq̄ pair, while the effective interaction used
in [20, 21, 22, 23, 24] operates, not only in spin singlet and isosinglet channel, but also in
spin triplet and isotriplet channel. It is known that the difference causes a discrepancy in
1Hayashigaki and Terasaki considers a possibility of the isotriplet state for Ds(2317) [16].
the meson mass spectrum [34]. Therefore it is an interesting problem to investigate the
isospin violation of the triquark by using the effective interaction in [20, 21, 22, 23, 24].
The content of this paper is as follows. In Section 2, the flavor representation of the
triquark, the III and OGE potential and the mass matrix are discussed. In Section 3, the
isospin mixing is investigated by considering the ud quark mass difference. In Section 4,
our discussion is summarized.
2 Quark model
In the tetraquark picture of the exotic Ds mesons, the triquark is considered as a bound
state composed by three light flavor quarks. The hamiltonian of the triquark is obtained
only in light flavors space, since the interaction between the light and heavy quarks
is suppressed in the OGE potential. This is also the case for the instanton induced
interaction, since the heavy quark has no zero mode and free from the instanton vacuum
[36, 37]. The flavor SU(3) multiplets of the triquark state qq̄q̄ are given as
3⊗ 3⊗ 3 = 3S ⊕ 3A ⊕ 6A ⊕ 15S.
We write the subscripts of S and A according to the symmetry under the exchange of two
anti-quarks. In Fig. 1, we show the weight diagrams of these multiplets. It is assumed
that all the quarks and anti-quarks occupy the lowest energy single particle orbital, the
s-wave orbital. In the following, we omit the subscripts in 6A and 15S for simplicity.
The exotic states reported in experiments have the strangeness S = +1. Then, the
isospin for each flavor multiplet is as follows; isosinglet for 3A, 3S and 15
, and isotriplet
for 6 and 15
. Here the isospin components of 15 are distinguished by the superscript. It
is straightforward to write down the flavor wavefunctions of these multiplets for S = +1.
isosinglet


|3A〉 = 12
u(s̄ū− ūs̄)− d(d̄s̄− s̄d̄)
|3S〉 = 12√2
2ss̄s̄+ u(s̄ū+ ūs̄) + d(d̄s̄+ s̄d̄)
|150〉 = 1
2ss̄s̄− u(s̄ū+ ūs̄)− d(d̄s̄+ s̄d̄)
isotriplet
|6〉 = 1
u(s̄ū− ūs̄) + d(d̄s̄− s̄d̄)
|151〉 = 1
u(s̄ū+ ūs̄)− d(d̄s̄+ s̄d̄)
In the following discussion, we consider only the S = +1 sector.
The triquark must belong to the color anti-triplet state, 3
S and 3
A, so that the
tetraquark is a color singlet state. Then, the spin and color combination of the tri-
quark is restricted by the Pauli principle. For example, the spin and color basis for the
flavor 3A and 6 states with the spin J = 1/2 is {|λX〉, |ρY 〉}. Here, λ and ρ stands for
the mixed states with λ- and ρ-symmetry in spin 1/2, and X and Y for color 3
S and 3
S S S
3 6 15
Figure 1: The weight diagram of the flavor SU(3) multiplets of the triquark qq̄q̄.
respectively. On the other hand, the basis for the flavor 3S and 15 states with spin 1/2 is
{|ρX〉, |λY 〉}. For spin J=3/2 state, we have |J=3/2 X〉 for 3A and 6, and |J =3/2 Y 〉
for 3S and 15.
Now we discuss the hamiltonian of the triquark. The instanton induced interaction
(III) has played very important role in the QCD vacuum in accompany with dynamical
chiral symmetry breaking [20, 21, 22, 23, 24]. It induces the Kobayashi-Kondo-Maskawa-
’t Hooft (KKMT) interaction [35, 36, 37], which is given as 2Nf point-like vertex with
the flavor anti-symmetric channel. In the quark model, the instanton effect has been
discussed in the non-relativistic limit in the KKMT interaction. The OGE potential is
also often used as an effective interaction [38]. Here we consider a hybrid model of the III
and OGE potentials [20, 21, 22, 23, 24]. The hamiltonian is
H = K + pIII
III +H
+ (1− pIII)VOGE +Mmass + Vconf , (2)
with the kinetic term K, the instanton induced interaction H
III (i = 2 and 3 for the two-
and three-body interactions), the OGE potential VOGE, the mass matrix Mmass and the
confinement potential Vconf . The parameter pIII controls the ratio of the III and the OGE
potentials. In the present discussion, we are interested in the isospin symmetry breaking,
and not involved with the absolute masses of the tetraquarks. Therefore, we pick up only
the III and OGE terms and the mass matrix;
H̃ = pIII
III +H
+ (1− pIII)VOGE +Mmass. (3)
Since we do not solve the quark confinement dynamically, we just use a quark wave
function from the harmonic oscillator potential with frequency ω.
Concerning the III potential, the three body force in the three quark state, q1q2q3, is
given in the flavor diagonal form as
III =
~λ1 ·~λ2 + ~λ2 ·~λ3 + ~λ3 ·~λ1
dabcλ
~σ1 ·~σ2~λ1 ·~λ2 + ~σ2 ·~σ3~λ2 ·~λ3 + ~σ3 ·~σ1~λ3 ·~λ1
0 −〈ψ̄ψ〉 mq ω αs
-0.2564 [GeV fm3] (0.25)3 [GeV3] 0.3837 [GeV] 0.5 [GeV] 1.319
Table 1: The parameter set from [21].
dabcλ
3 (~σ1 ·~σ2 + ~σ2 ·~σ3 + ~σ3 ·~σ1)
ǫijkσ
3fabcλ
δ(3)(~r1 − ~r2)δ(3)(~r2 − ~r3),
with a coupling constant V
0 and the delta functions as a point-like three body interaction.
We can deduce the two body instanton induced force,
III =
~λi ·~λj +
~σi ·~σj~λi ·~λj
δ(3)(~ri − ~rj),
using the quark condensate 〈ψ̄ψ〉, where the coupling constant V (2)0 is given as
〈ψ̄ψ〉V (3)0 .
The interactions in the q1q̄2q̄3 state are also obtained in a straightforward way.
The OGE potential between the q1q2 pair is given as
VOGE = 4παS
~λ1 ·~λ2
− ~σ1 ·~σ2
6m1m2
with a coupling constant αS. The first term is the electric interaction, and the second the
magnetic interaction with spin dependence. However, we neglect the electric interaction,
since in general it is sufficiently small as compared with the magnetic interaction. It should
be noted that the magnetic interaction is switched off with a suppression of 1/mQ for the
heavy-light quark pair (Qq) in the limit of the heavy mass. Therefore, it is understood
that the triquark qq̄q̄ may exist as a compound unit in the tetraquark structure. As a
summary, our interaction is sketched in Fig. 2. The parameter set in our interaction [21]
is summarized in Table 1.
From the III and OGE potentials, the energy spectrum of the triquark is obtained
in the following way. By using the basis of the spin and color, {|λX〉, |ρY 〉, |ρX〉, |λY 〉},
we obtain the hamiltonian in matrix forms for flavor 3A, 3S, 6 and 15 representations,
respectively.
First we consider the III potential. For the flavor 3A and 3S states, the hamiltonian
is given in the basis {|λX〉, |ρY 〉, |ρX〉, |λY 〉} by
HIII(3) =

0 3 −3

0 I2 +

0 0 0 0
0 0 0 0

0 I3, (4)
(c) qq 
(d) qq 
(e) qq (annihilation) 
(a) two-body (qq) 
(b) three-body (qqq) 
Figure 2: The diagram contributions for the III and OGE potentials. III: (a) the two-body
interaction for qq̄ and (b) the three body interaction for qq̄q̄. OGE: (c) qq̄, (d) q̄q̄ and (e) qq̄
(annihilation).
where I2 and I3 are the expectation values of the delta function for the point-like inter-
action,
I2 = 〈Ψ|δ(3)(r1 − r2)|Ψ〉 =
for the two-body interaction, and
I3 = 〈Ψ|δ(3)(r1 − r2)δ(3)(r2 − r3)|Ψ〉 =
for the three-body interaction with the triquark spatial wavefunction Ψ. It should be
mentioned that the 3A and 3S states are mixed due to the off-diagonal element in the two-
body interaction in the III potential, since {|λX〉, |ρY 〉} belongs to 3A and {|ρX〉, |λY 〉}
to 3S. Therefore, we may denote the mixed state as 3 in the following discussion. In the
similar way, for the flavor 6 state, the basis {|λX〉, |ρY 〉} gives the matrix
HIII(6) =
0 I2 +
0 I3, (7)
and for the flavor 15 state, the basis {|ρX〉, |λY 〉} gives
HIII(15) =
0 I2. (8)
Second we consider the OGE potential. For the 3 state, we obtain the matrix in the
basis of {|λX〉, |ρY 〉, |ρX〉, |λY 〉},
VOGE(3) =


I2. (9)
Furthermore, for the 6 state, we obtain in the basis {|λX〉, |ρY 〉}
VOGE(6) =
I2, (10)
and for the 15 state
VOGE(15) =
I2, (11)
in the basis {|ρX〉, |λY 〉}.
The mass differences among u, d and s quarks induce mixings between the flavor
representations. In the basis of the flavor representation, {|3A〉, |3S〉, |15
0〉, |6〉, |151〉}, we
easily obtain the mass part of the hamiltonian for S = +1 sector, as
Mmass =

mu +md +ms 0 0 mu −md 0
0 mu+md
+ 2ms −mu+md2 +ms 0
mu−md√
0 −mu+md
+ms 0 0 −mu−md√2
mu −md 0 0 mu +md +ms 0
0 mu−md√
−mu−md√

.(12)
The diagonal elements are isosinglet and isotriplet components, while the off-diagonal
elements induce mixings between them. Note that the flavor representations with the
same symmetry (A or S) are mixed. The 3A and 15
states are mixed with each other
by the SU(3) symmetry breaking ( mu = md < ms). We also note that the isosinglet
states (3A, 3S, 15
) and the isotriplet states (6, 15
) are also mixed due to the isospin
symmetry breaking (mu < md)
2. We consider this interaction as a driving force for the
isospin symmetry breaking in the next section. It should be noted that the Coulomb or
electromagnetic interaction may also break isospin symmetry, which is not considered in
this study.
3 Isospin mixing
In general, the u − d quark mass difference is sufficiently small as compared with the
energy splitting between the isosinglet and isotriplet states, and the isospin breaking can
2In the works in [31, 32, 33], the 3A and 6 states are mixed due to the mu = md 6= ms. As long as
the isospin symmetry is not violated, however, we have no mixing between the 3A and 6 states.
be neglected. However, in the triquark, we see that the isosinglet and isotriplet states
sometimes happen to be degenerate and thus a large isospin mixing can occur. In this
section, we investigate the mixing of the isosinglet and isotriplet states. For this purpose,
we calculate the eigenenergies, E, of the hamiltonian (3). We choose the s quark mass
ms = 0.48 GeV and the strength of the harmonic potential ω = 0.50 GeV in the following
discussion. We take the parameter pIII as a free parameter.
We present the binding energy spectrum of the triquark, ∆E = E − (mu +md +ms),
for the OGE (pIII = 0) and III (pIII = 1) potentials in Fig. 3. The isosinglet and isotriplet
states are shown by the solid and dashed lines, respectively. As the J = 3/2 states are
heavier than J = 1/2, in the following discussion, we pay attention to the ground states
with the spin J=1/2, the 3 and 6 multiplets.
Let us see the result by the III potential. In SU(3) symmetric case, the ground state
is the 3 state, which contains mainly the 3A component rather than the 3S component.
On the other hand, the 3S component is mixed in the excited state in the 3 state. Now
we break the SU(3) symmetry with keeping the isospin symmetry; mu = md < ms. The
ground state is still the 3 state, and the first excited state is the 15
state, followed by
the 15
and 6 states. The splitting between the 15
and 15
states makes the former
lifted up as compared with the latter. This splitting comes from the fact that the mass
matrix (12) mixes the 3S and 15
states. The same mixing pushes the 3 upward.
On the other hand, in the OGE potential, the ground state is the 6 state, followed by
the 3, 15
, and 15
states. It should be noticed that the flavor multiplets are different
in the III and the OGE potentials. Especially the change of the ground state flavor is
important for the isospin symmetry breaking as we see below.
The reason that the 6 state is the ground state in the OGE can be understood by
examining the annihilation diagram in Fig. 2(e). It vanishes for the usual color singlet
meson qq̄, since the gluon (g) contained in the process qq̄ → g → qq̄ is a color octet state.
In the triquark, however, the annihilation diagram does not vanish. This is because the
gq̄ state contained in the process qq̄q̄ → gq̄ → qq̄q̄ remains color anti-triplet 3c due to the
color decomposition,
c ⊗ 3c = 3c ⊕ 6c ⊕ 15c.
Note that the initial and final qq̄q̄ states are also color anti-triplet. The annihilation term
increases the energy of the flavor 3 state, while it does not operate for the 6 state (see Eq.
(1) ). Consequently, the 3 state is about 50 MeV above the 6 state in the OGE potential.
Let us return to the discussion of the isospin symmetry breaking. We recall that the
3 state is isosinglet and the 6 state is isotriplet. Thus the mixing of the flavor multiplets
are directly related to the isospin mixing. Explicitly, we plot the binding energies of the
flavor multiplets as functions of the parameter pIII in Fig. 4. The solid lines indicate the
isosinglet states, and the dashed lines the isotriplet states. We find that the isosinglet
and isotriplet states become degenerate at A (pIII = 0.18) and B (pIII = 0.82).
Now let us introduce the isospin symmetry breaking, namely the ud quark mass differ-
ence, ∆m = md −mu ≃ 0.005 GeV [39]. The ∆m is comparable to the energy difference
between the isosinglet and isotriplet states at A and B. There, the two degenerate states
ΔE [GeV]
(p  =0)
(p  =1)
SU(3)
 breaking
SU(3)
 symmetry
SU(3)
 breaking
SU(3)
 symmetry
III III
isosinglet
isotriplet
Figure 3: The binding energies of the triquarks with J = 1/2 for the OGE (pIII = 0) and III
(pIII = 1) potentials.
0 0.5 1
I=0 (3bar, 15bar)
I=1 (6)
I=1 (15bar)
Figure 4: The binding energies of the various flavor multiplets with SU(3) breaking as functions
of the parameter pIII . The bold-solid line indicates the isosinglet (3 and 15
) states, the bold-
dashed line isotriplet (6), and the thin-dashed line the isotriplet (15
) states. Cf. Fig. 3.
0 0.1 0.2 0.3 0.4
uubar
ddbar
ssbar
0.6 0.7 0.8 0.9 1
uubar
ddbar
ssbar
Figure 5: The ratios of the isosinglet (bold-solid line) and isotriplet (bold-dashed line) states
as functions of the parameter pIII . (a) and (b) corresponds to the state A and B, respectively,
in Fig. 4. The ratios of the uū (thin-solid line), dd̄ (thin-dashed line) and ss̄ (thin-dot-dashed
line) are also shown.
will split into two isospin mixed states which are orthogonal to each other. Here we choose
one state at A. The ratios of isosinglet and isotriplet components are plotted as functions
of the parameter pIII in Fig. 5(a). In the range of 0.16 < pIII < 0.20, we see a rapid
change of the isosinglet (bold-solid line) and the isotriplet (bold-dashed line) components,
hence the isospin is strongly mixed. In the same way at B, we also see an isospin mixing
at pIII = 0.82 as shown in Fig. 5(b). However, in contrast to the case A, the isospin
mixing at B occurs in a small range of the parameter pIII . This is understood from the
mass matrix (12). At A, the isosinglet state is almost the 3A multiplet, while the isotriplet
state is purely the 6 multiplet (see Fig.3). The mass matrix (12) induces the 3A and 6
multiplet mixing, namely the isospin violation, by mu−md. On the other hand, at B, the
isosinglet state is changed to be the 15
state, while the isotriplet state is the same. In the
mass matrix (12), however, there is no direct mixing between the 15
and 6 multiplets.
They are mixed indirectly through the multi-step mixings of the 6 − 3A, 3A − 3S, and
3S − 15
. Therefore the isospin mixing at B is suppressed as compared to that at A.
Here we recall the isospin violation in experimental observations. Maiani et al. con-
sider Ds(2632) as the cs̄dd̄ state, which is an isospin mixed state [17]. In our analysis, the
isospin mixing at A induces a mixing between the isosinglet (mostly 3A) and isotriplet (6)
states. Hence, from Eq. (1), the mixed wavefunction, |3A〉 − |6〉 = −d(d̄s̄− s̄d̄), contains
only the dd̄ component. On the other hand, at B, there is a mixing between the isosinglet
(mostly 15
) and the isotriplet (6) states. There, from Eq. (1), the mixed wavefunction
|150〉 − |6〉 contains both of the uū and dd̄ components with the same fraction. Conse-
quently, we see that the isospin mixed states, cs̄uū and cs̄dd̄, become separate eigenstates
by the 3− 6 mixing rather than the 150 − 6 mixing.
We also understand this result explicitly by looking at the fraction of uū, dd̄ and ss̄
components at A and B in Fig. 5(a) and (b), respectively. In Fig. 5(a), the dd̄ fraction
(thin-dashed line) is overwhelming as compared with the uū fraction (thin-solid line)
around pIII = 0.18. In contrast, in Fig. 5(b), the fraction of the uū and dd̄ components
are almost the same at pIII = 0.82. Therefore, the isospin mixed state at A gives the
cs̄dd̄, while the state at B does not. Thus, the discussion by Maiani et al. in [17] is proven
to be possible as the the 3− 6 mixing.
So far, we have discussed the isospin mixing by using the isospin basis of isosinglet
and isotriplet. However, the isospin mixing is also investigated by basis {uū, dd̄}. Then
the hamiltonian is generally given by
(uū dd̄
uū m δ
dd̄ δ m+ 2∆m
. (13)
The uū and dd̄ are eigenstates of this hamiltonian, if the flavor mixing term δ is much
smaller than ∆m, and only the diagonal component is dominant. However, in general, δ
is in the order of hundred MeV in the vacuum as we see the mass splitting of π − η.
For the triquark with J = 1/2, due to the combination of spin and color, we have four
uū-like states and also four dd̄-like states. In this basis, the hamiltonian is given by
uū dd̄

m1 0 δ11 δ12 δ13 δ14
m2 δ21 δ22 δ23 δ24
m3 δ31 δ32 δ33 δ34
0 m4 δ41 δ42 δ43 δ44
δ11 δ12 δ13 δ14 m1+2∆m 0
δ21 δ22 δ23 δ24 m2+2∆m
δ31 δ32 δ33 δ34 m3+2∆m
δ41 δ42 δ43 δ44 0 m4+2∆m

where the diagonal uū−uū and dd̄−dd̄ parts are diagonalized in the spin and color spaces.
The diagonalized energy, m1, m2, m3 and m4, are plotted as functions of the parameter
pIII in Fig. 6(a). If the flavor mixing strength δij (i, j = 1, · · · , 4) are sufficiently small,
the lowest uū- and dd̄-like states become eigenstates. As shown in Fig. 6(b), δ11 is so
small as compared with ∆m around pIII = 0.18. There, the eigenstate become uū- and
dd̄-like states, hence the isosinglet and isotriplets states are ideally mixed. This result is
consistent with our discussion that the isospin mixing is caused by the 3−6mixing around
pIII = 0.18. It should be noted that the contribution from the higher states is suppressed
since the mixing is in the order of δij/(mk−m1) ≃ 0.1 (k ≥ 2) in the perturbation theory.
Therefore the first order perturbation is sufficient for the present discussion.
Lastly, we discuss the parameter dependence of the isospin mixing. We employ several
free parameters, the s quark mass ms, the harmonic oscillator potential frequency ω and
the parameter pIII for the OGE and III potentials. They may have some uncertainty due
to the lack of the experimental information. However, one sees that the results are not
0 0.5 1
0 0.1 0.2 0.3 0.4
Figure 6: (a) The diagonal components of the uū − dd̄ matrix as functions of the parameter
pIII . (b) δ11 as a function of pIII . Note the energy unit is given by MeV in (b). See the text.
modified qualitatively by parameter change. As an example, we plot the size parameter
b = 1/
mqω of the triquark wavefunction, which causes the 3 − 6 mixing at A, as a
function of the parameter pIII for ms = 0.48 GeV and 0.58 GeV. We see that the b comes
within a reasonable range 0.4 < b < 0.6 fm, and is not far from b = 0.5 fm [21]. This
range is little affected by ms.
Concerning the range of the parameter pIII , the obtained value pIII = 0.18 is smaller
than the conventionally used value pIII ≃ 0.4 in hadron spectroscopy [21]. This observa-
tion indicates that the OGE is more dominant than the instanton induced interaction in
tetraquarks. It is noticed that the value pIII = 0.18 is not obtained dynamically, since the
quark wave function is assumed to be Gaussian. The present study suggests that there
would exist an essential mechanism to choose such pIII in charmed tetraquark.
When the linear potential is used as a confinement potential, the quark wave function
is modified from that of the harmonic oscillator potential, and the absolute values of
the OGE and the instanton induced interaction are also modified. However, the ratio of
both interactions is not changed, since both of the potentials are point-like interactions.
In the present discussion, the isospin symmetry breaking is induced by the ratio of two
interactions. Therefore our conclusion is not modified qualitatively.
4 Conclusion
Possibility of isospin violation in the Ds tetraquark systems is examined in this paper.
Tetraquarks are candidates of the exotic Ds mesons recently reported in experiment. We
consider the energy spectrum of the triquark by using the non-relativistic quark model
with the instanton induced interaction and the one-gluon exchange potentials. With
taking the SU(3) symmetry breaking into account for S = +1 sector, we show that the
flavor 3 (isosinglet) and the flavor 6 (isotriplet) representations form the ground states.
Considering the isospin symmetry breaking by the quark mass difference, mu < md, it is
0 0.5 1
ms=0.48 [GeV]
ms=0.58 [GeV]
Figure 7: The b− pIII relation for the 3− 6 mixing. ms = 0.48 GeV (solid line) and 0.58 GeV
(dashed line). See the text.
shown that the 3 (isosinglet) and 6 (isotriplet) states may be mixed strongly with some
range of the parameter pIII . There the isosinglet and the isotriplet states are ideally
mixed, and one of the eigenstates is dominated by the dd̄ component. This result is
also investigated by looking at the off-diagonal components in the uū − dd̄ matrix. Our
conclusion supports the discussion given in [17, 18].
How do we experimentally confirm the picture given in this paper? The present
mechanism of the isospin symmetry violation relies on the suppression of the flavor mixing
interaction. Thus, at the ideal (maximal) mixing, the uū- and dd̄-like states are split by
the diagonal part of the mass matrix, namely by 2∆m ∼ 10 MeV. Therefore the two states
are expected to come close to each other. So far, due to the experimental restriction, only
a few charged decay modes are observed, and they suggest a dd̄-like state, D+s (2630/cs̄dd̄),
where its main decay mode is D+s (2630) → Dsη, while D+s (2630) → D0K+ is suppressed.
The corresponding uū-like state will show different decay patterns. Therefore careful
analyses of different charged modes of decays will reveal the nature of the isospin breaking.
In particular, the decays into Dsπ
0 and D+K0 are two interesting modes.
The present study suggests the possibility of the triquark à la “color non-singlet
baryon”, which is a color non-singlet particle composed by three quarks. Although the
triquark itself cannot exist asymptotically, it may appear as an effective degree of freedom
in the exotic heavy mesons in heavy quark mass limit. It is considered in general that
the color non-singlet light quark systems may exist by color neutralization with heavy
quark spectator [30]. The triquark is a possible candidate among the color non-singlet
quark systems, which can be examined by studying the tetraquark structure of exotic
open charm mesons. The triquark would be also an interesting object in the lattice QCD
simulation. Furthermore the triquark may be a relevant degree of freedom as a color non-
singlet compound particle in the deconfinement phase such as the quark-gluon plasma and
the quark matter. In order to understand such states in many aspects, it is important to
study several properties, such as masses, decay widths and so forth.
Acknowledgment
We express our thanks to Dr. T. Shinozaki and Prof. S. Takeuchi for discussions. This
work is supported by a Grant-in-Aid for Scientific Research for Priority Areas, MEXT
(Ministry of Education, Culture, Sports, Science and Technology) with No. 17070002.
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	Introduction
	Quark model
	Isospin mixing
	Conclusion
