We discuss possibility of the existence of tetraquark states made of four
quarks in the string (flux tube) model. The new particle is composed of a
diquark and an anti-diquark which are connected by a color flux. It is shown
that the vibrational and rotational excited states of the string explain some
non-$\bar{q}q$ mesons observed experimentally. Moreover we discuss the decay
widths of such tetraquarks with the use of the Schwinger mechanism.Comment: 5 pages, 6 figure

Fukutome, Takahiko

Iwasaki, Masaharu

English

arXiv

Kochi University Repository

PHYSICAL REVIEW D 72, 094016 (2005)
Tetraquark particle in the string model
Masaharu Iwasaki* and Takahiko Fukutome
Department of Physics, Kochi University, Kochi 780-8520, Japan
(Received 15 November 2004; revised manuscript received 24 May 2005; published 18 November 2005)*Electronic
1550-7998=20We discuss the possibility of the existence of tetraquark states made of four quarks in the string (flux
tube) model. The new particle is composed of a diquark and an antidiquark which are connected by a color
flux. It is shown that the vibrational and rotational excited states of the string explain some non- qq mesons
observed experimentally. Moreover we discuss the decay widths of such tetraquarks with the use of the
Schwinger mechanism.
DOI: 10.1103/PhysRevD.72.094016 PACS numbers: 12.39.Ki, 12.40.Yx, 13.25.2k, 13.30.EgI. INTRODUCTION
It is well known that there are only two kinds of hadrons
in Nature: mesons and baryons. The former are made of a
quark and an antiquark and the latter made of three quarks.
Their existence is explained by quantum chromodynamics
(QCD) [1] that is the fundamental theory for hadron phys-
ics. There is, however, a possibility of the existence of
other type hadrons, such as four-quark particles. In fact a
possible four-quark state has been suggested experimen-
tally by the several groups [2].
In this paper we discuss the possible existence of a four-
quark state which we call a ‘‘tetraquark’’; it is made up of
two quarks and two antiquarks. Since Jaffe predicted such
hadrons with the use of the MIT bag model [3,4], many
non- qq mesons have been discussed by many authors [5–
11]. Particularly it is conjectured that the nonet of light
scalar mesons [f0600, f0980, a0980, 800?] with
masses below 1 GeV seem to fit a qq q q description.
Instead a heavier scalar nonet above 1 GeV seems to be
largely qq configuration. The light scalars seem to have a
significant component of qq q q and the pairs of quarks
(and antiquarks) are separately correlated in a flavor, color
and spin antisymmetric state. A diquark and an antidiquark
neatly couple together to give a nonet of color-singlet
scalar mesons. There is also evidence for strong diquark
correlations in the baryon spectrum [12].
We will consider a possibility that this strong diquark
correlation leads to a molecularlike configuration, qq 
 q q with the use of the chemical bondlike expression. To
this end, we will study the four-quark system by using the
hadron string model or the flux tube model. According to
the model, mesons are made up of a quark and an antiquark
which are connected by the classical gluon field [13]. The
baryons are also made up of a quark and a diquark con-
nected by the same flux [14,15]. Using this simple model,
F. Takagi and one of the present authors (M.I.) have shown
that many excited states of hadron observed experimen-
tally can be identified as the vibrational and rotational
excited states of this one-dimensional string [16,17].address: miwasaki@cc.kochi-u.ac.jp
05=72(9)=094016(5)$23.00 094016Since any hadron must be a color-singlet, the only
possible four-quark state is a qq q q state. There are here
two configurations: (a)  qq   qq and (b) qq   q q.
We suppose that other configurations q  q q q and
 q  qq q are not so stable. Assuming the one-gluon
exchange interaction, the q q system is attractive in the
color-singlet state (meson) so that the system (a) probably
decays into two mesons at once. On the other hand qq
( q q ) system is attractive in the color antitriplet (3 bar)
state, which means that it is equivalent to q (q) in the color
SU(3) space. That is the system (b) is the same as the
meson as far as the color degree of freedom. Therefore a
four-quark state considered in this paper is assumed to be
composed of a diquark and an antidiquark connected by the
same gluon flux as the meson.
It should be emphasized here that our approach is differ-
ent from that in Jaffe’s papers. According to them, the
nonet of light scalar is described by the s-wave qq q q
sector in the MIT bag model; all the quarks are in the
same s-state in the bag potential. On the other hand, a
scalar in our model is described by the s-state of the
diquark and antidiquark where each quark is also s-state.
The latter s-state is quite different from the former one.
The wave function of the quark would be a superposition of
many states of the higher angular momenta in the MIT bag
potential.II. A FLUX TUBE MODEL
According to [16], our system is described by the rela-
tive distance r between the diquark and the antidiquark and
the rotation angle  of the string. The geometry is shown in
Fig. 1 where their masses is denoted by M1 and M2. We
take the origin as the center of mass. Since our discussion is
restricted to the nonflavored mesons, we assume M1 
M2  M. Then the Lagrangian of this system is given by
Lr; _r; !  2M

1 v2
p
 Vr;!; (1)
where v2   _r2  r2!2=4 is the square velocity of each
diquark. The first term represents the action of the diquark
and the antidiquark and the second one the potential energy
of the system which is represented by-1 © 2005 The American Physical Society
FIG. 1. The schematic picture of the hadron string.
IWASAKI AND FUKUTOME PHYSICAL REVIEW D 72, 094016 (2005)Vr; !  2
Z r=2
0
a

1 r2!2
p
dr e
r
: (2)
The first term denotes the energy of the gluon flux with the
string tension a (the energy per unit length) [18] and
1 r2!2
p
comes from the Lorentz contraction of the
transversal motion. The second one represents the one-
gluon exchange potential with the coupling constant e.
This expression for the potential is known as Cornel po-
tential and confirmed by the Lattice QCD simulation
[19,20]. It should be noted that this Lagrangian is also
valid for the qq system (ordinary mesons) because their
color charges are the same as those of the qq q q one.
The Hamiltonian of our system is defined by H  _rp
!l L where p  @L=@ _r and l  @L=@ _. Then it is
written as
Hp; l; r  2M
1 v2
p 
Z r=2
0
2a
1 r2!2
p dr e
r
: (3)
where the factor

1 r2!2
p
in the right-hand side repre-
sents the Lorentz contraction along the transverse direction
of the string. The excited states of the system are described
by the Schro¨dinger equation, Hp; l; r r;   E r; .
Carrying on the separation of variables   =r expim,
this equation is reduced to
H

i @
@r
;m; r

r  Er; (4)
where l is the angular momentum quantized as m 
0; 1; 2; 3;    . This equation can be solved using the
WKB approximation and the excited energy E is obtained
by the Bohr-Sommerfeld formula,
I
prdr  n 1=2h: n  0; 1; 2; 3;    (5)
The right-hand side should be replaced by n 3=4h
when m  0. The full derivation of the formula is ex-
plained in Ref. [16].
Here let us discuss the qualitative properties of the mass
spectra (neglect the masses and the one-gluon exchange
term for convenience). In the case of m  0 (vibrational
motion), the energy becomes E  2p ar  ar0 (r0 is094016the maximum length). This leads to Hpdr  1=2ar20 
2n. Thus we have
E2  4an: (6)
In the case of n  0 (rotational motion), the energy is E 
2m=r ar if the kinetic energy is approximated by the
centrifugal potential. This is minimized by r  2m=ap so
that E  2ar. These relations lead to
E2  8am: (7)
In both cases, it is concluded that the energy squared E2 is
proportional to the quantum numbers n or m and the ratio
of the proportional constants is approximately :2.
Next let us consider the decay width of our system. The
possible decay process that we consider is a pair produc-
tion of qq inside the flux tube which is known as Schwinger
mechanism [21–24]. The probability of the pair production
in a unit space-time volume in the tube is given by
w  a
2
43
X
q
X1
n1
1
n2
exp

nm
2
q
a

: (8)
where a is the string tension and
P
q indicates a summation
over all quark flavors with mass mq (q  u; d; s) [22–24].
The probability of the decay (pair production) in the time
interval dt is given by
dW 
Z
j r j2 wvrdrdt; (9)
where vr denotes the volume of the string with the length
r and is given by
vr  2
Z r=2
0

1 r2!2
p
dr: (10)
In the right-hand side,  denotes the area of the cross
section of the tube in the rotating coordinates system.
The decay width of our system is given by
  dW
dt

Z
jrj2wvrdr; (11)
This formula gives the decay widths of the excited hadrons
due to the quark pair production. As is shown in this
formula, the decay width is proportional to the length of
the string. This means that it increases as the excited
energy increases.
III. NUMERICAL RESULTS
We are in a position to carry out the numerical calcu-
lations of our tetraquark particles. Our string model has
two components, the color flux and the diquark. The for-
mer is characterized by the string tension a and the cross
sectional area  and the latter is concerned to the diquark
mass. The coupling constant e will be used to be e  0:4,
which is a usual value [19]. The parameters concerning to
the color flux should be taken as the same values as those of-2
FIG. 3. The vibrational excited states of the tetraquark. The
quantum number of the vibrational motion is denoted by n.
TETRAQUARK PARTICLE IN THE STRING MODEL PHYSICAL REVIEW D 72, 094016 (2005)
q q system, because a diquark (antidiquark) has the
same color charge as that of an antiquark (quark). These
parameters have been determined so as to reproduce the
experimental mass and decay width of 770.
When we take a  0:11 GeV2 and (M1  M2 
10 MeV), the calculated vibrational spectrum of the rho
meson is shown in Fig. 2. In the case of m  0, the interval
of the integral (5) is 0< r < r0 so that the integral di-
verges. Therefore the cutoff parameter   0:01 fm  <
r < r0 has been introduced so as to reproduce the mass of
770. Its decay width (  150 MeV) determines the
other parameter,   75 GeV2. We use these parameters
in later calculations hereafter.
As for the diquark mass, we take the most simple
diquark which is composed of u and d quarks. In other
words we restrict ourselves to the flavor SU(2). In this case,
the most stable diquark is color antitriplet, flavor-singlet
and spin-singlet taking into account the Fermi statistics.
Since the nucleon is made up with this diquark and the
remaining quark, our diquark mass is determined by the
nucleon mass N(939) as is done in Ref. [16]: M 
300 MeV.
First it is adequate to discuss the possible quantum
numbers of our tetraquarks, which has configuration
qq   q q. Since the diquark (antidiquark) is flavor-
singlet and spin-singlet, it is isoscalar (I  0) and has not
spin angular momentum. The total angular momentum J is
nothing but the orbital one: J  m. The G-parity operation
defined by G  CeiI2 is the same as that of the ordinary
parity P, because the charge conjugation C is equivalent to
P in our system. Therefore we should seek mesons with
IGJPC  0	J		.
Let us discuss the vibrational excited states with the
angular momentum m  0. The square of mass is drawn
as a function of the quantum number n in the Fig. 3. The
closed circles in the figure represent the corresponding
particles with 00 tentatively. They are f1110,
f01500, f02020, f02200 [25]. The head state
f(1110) is 0even and its spin J is not determined
except for J  even integer. It is seen our simple modelFIG. 2. The vibrational excited states of the rho meson. The
quantum number of the vibrational motion is denoted by n.
094016reproduces such particles fairly well. The trajectory is an
almost straight line as is shown in the Eq. (6).
Next let us consider the rotational excited states with
n  0, which is called Regge trajectory. In the Fig. 4, the
square of mass is shown as a function of the angular
momentum. As discussed above, the quantum numbers of
the Regge trajectory should be 00, 01,
02, 03,    . So the closed circles represent
the corresponding particles tentatively: f1110,!11420,
f21640, !31945, f42050, !5?, f62510. The candi-
date for !5 has not been observed yet. We see that our
model explains such particles fairly well. The deviation
from the straight line is due to the diquark mass.
Next we go over to the discussion on the decay width of
the tetraquark. Here it is interesting to notice that our
tetraquarks with E< 2 GeV can not decay with the pro-
cess of the pair production by virtue of the conservation of
energy. Because the total energy of the decay products (a
baryon and an antibaryon) must be larger than about
2 GeV. Therefore it is predicted that the decay widths of
f1110, f01500 are suppressed and those of f02020
and f02200 are enhanced. The results are plotted as
circles on the M; plane [25] as shown in Fig. 5. The
decay mode to baryon-antibaryon has not been observed
yet in the case of f02020 [26,27]. The experiments inFIG. 4. The rotational excited states of the tetraquark. The
orbital angular momentum is denoted by m
-3
FIG. 5. The masses and decay widths of all the unflavored
mesons observed experimentally are plotted by the open circles.
The filled ones represent the candidates for the vibrational
excited states of the tetraquark. The corresponding calculated
states are denoted by the triangles.
FIG. 6. The vibrational excited states of the q qmesons. The
quantum number of the vibrational motion is denoted by n.
IWASAKI AND FUKUTOME PHYSICAL REVIEW D 72, 094016 (2005)Fig. 5 is focused on the process pp! pf4ps. We
consider that even if a baryon-antibaryon pair is created
in the process, the antibaryon would interact at once the
other baryon and decay into pions. As for the decay of
f02200, it is seen in the experiment pp! ~~ where
only the ~~ mode is selected [28]. Although it is not
detected in the experiments, we conjecture a resonance
reaction of pp! pp. This resonance would support the
existence of the tetraquark considered in our model.
The calculated decay widths are shown by the triangles.
The black circles represent f1110, f01500, f02020,
f02200 and the open ones all the other unflavored mesons
observed experimentally. The enhancement of the decay
widths of f02020, f02200 may be explained by the open
of the decay channel due to the pair production. The decay
widths of f1110 and f01500 are relatively small and
caused by the other decay modes. Of course it is not
adequate to expect quantitative discussions on the experi-
mental values by our model. Because our model is over-
simplified and the data on the decay widths have large
ambiguity. Here we are satisfied with that the calculated
decay widths are the same order of magnitude as the
experimental data.IV. DISCUSSIONS AND SUMMARY
In the previous section, we have studied the scalar
mesons above 1 GeV from the standpoint of the qq q q
model. There are, however, the other scalar mesons not
described by the present model. In fact they seem to be
f01370 and f01710. There is a possibility that they are
described by the usual q q model. We will investigate
such scalar mesons using the simple q q model in this
section.
Let us consider a meson composed of q and q connected
by the color flux. If the meson is p-wave, we have the094016scalar meson, because the intrinsic parity of q is minus.
The mass spectrum of this system is calculated in the same
manner as was done in the previous section. In particular
we restrict ourselves to the case ofm  1 (p wave). The
vibrational excited states are shown as a function of n in
Fig. 6. The closed circles in the figure represent the corre-
sponding particles f01370 and f01710 tentatively.
Therefore it is suggested that these particles are composed
of a quark-antiquark in the p-wave state.
What is a discrimination between the qq q q and q
q configurations? One possible clue is the decay width.
That of the four-quark particle is suppressed for the con-
servation of energy as mentioned before. In fact the decay
width of f01370 is larger than 1 GeV. However that of
f01710 is not so large, that is 140 MeV, which is larger
than those of f1110 and f01500 [25].
In conclusion, we have studied possibility of the exis-
tence of the tetraquark states which are composed of a
diquark and an antidiquark connected by a color flux. The
vibrational and rotational excited states of the string are
calculated with the use of the WKB approximation. It is
suggested that they may be identified as the observed
mesons. Moreover the decay widths of the tetraquarks
are calculated using the Schwinger mechanism. They are
suppressed by the conservation of energy, which is con-
sistent with the experimental results. It is a very interesting
problem to study the possibility of the tetraquark.ACKNOWLEDGMENTS
The authors would like to thank Professor F. Takagi at
Ishinomaki Senshu University and Professor K. Yamawaki
at Nagoya University for helpful discussions. They also
thank the Yukawa Institute for Theoretical Physics at
Kyoto University. Discussions during the YITP workshop
YITP-W-04-07 on ‘‘Thermal Quantum Field Theories and
Their Applications’’ were useful to complete this work.-4
TETRAQUARK PARTICLE IN THE STRING MODEL PHYSICAL REVIEW D 72, 094016 (2005)
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Tetraquark particle in the string model

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