Axial Vector Tetraquark with Two s-quarks
Axial Vector Tetraquark with Two s-quarks
Yoshiko Kanada-En’yo1, Osamu Morimatsu2 and Tetsuo Nishikawa3
1 Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606–8502,
Japan
2 Institute of Particle and Nuclear Studies, High Energy Accelerator Research
Organization, Ibaraki 305-0801, Japan
3 Department of Physics, Tokyo Institute of Technology, 2-12-1, Oh-Okayama,
Meguro, Tokyo 152-8551, Japan
Possibility of an axial vector isoscalar tetraquark uds̄s̄ is discussed. If a f1 meson in the
mass region 1.4 − 1.5 GeV consists of four quarks nsn̄s̄, the mass of the isoscalar uds̄s̄(ϑ+-
meson) state with JP = 1+ is expected to be lower than that of the f1 meson. Within a
flux-tube quark model, a possible resonant state of uds̄s̄(JP = 1+) is suggested to appear
at ∼ 1.4 GeV with the width O(20 ∼ 50) MeV. We propose that the ϑ+-meson is the
good candidate for the tetraquark search, which would be observed in the K+K+π− decay
channel.
The possibility of multiquark states has been discussed for a long time.1)–10)
In particularly, the possible qqq̄q̄ states have been suggested in many theoretical
efforts to understand light scalar mesons(for example, Refs.[1,3,11]). The 4q states
were proposed in the description of f0(600) and f0(980), where the strong attraction
between (qq)3̄ and (q̄q̄)3 play an important role.
1), 11) Here, (qq)3̄ and (q̄q̄)3 denote the
color-anti-triplet quark pair and the color-triplet anti-quark pair, respectively. On
the other hand, the KK molecule states were suggested to understand the properties
of f0(980) and a0(980).
3) Even if the 4q components are dominant in a certain
meson with the conventional flavor, it is difficult to find an direct evidence of the 4q
components due to the mixing with the conventional qq̄ state via the annihilation of
qq̄ pairs. Our main interest here is the possibility of “tetraquark” states which has
the minimal 4-quark content.
The recent observation of DsJ(2317)
12) and the reports of the pentaquark baryon
Θ+(uudds̄),13) revived the motivation of the experimental and theoretical researches
on multiquarks in hadron physics, though the existence of Θ+ yet to be well estab-
lished. One of the striking characteristics of the Θ+ is its narrow width. For the
theoretical interpretation of the narrow Θ+ state, the possibility of the spin-parity
JP = 1/2+ and JP = 3/2− have been discussed by many groups.10), 14)–19) The
unnatural spin and parity is a key of the suppressed width for the lowest decay
channel.
Now we turn to the discussion on the possibility of the tetraquarks. If we accept
the interpretation of the pentaquark as the (ud)3̄(ud)3̄q̄ state based on the diquark
picture by Jaffe and Wilczek, it is natural to expect that a tetraquark with the uds̄s̄
content may exist at the nearly same energy region by replacing a ud-diquark with s̄
quark. Firstly, one should consider the states with unnatural spin and parity, which
cannot decay into two light hadrons (pseudo scalar mesons) in the S-wave channel.
typeset using PTPTEX.cls 〈Ver.0.9〉
http://arxiv.org/abs/0704.1856v1
2 Letters
Second, the exotic color configurations (qq)3̄(q̄q̄)3 would be essential to stabilize
the exotic hadrons. Then, we propose a JP = 1+ uds̄s̄ state with the (qq)3̄(q̄q̄)3
configuration as the candidate of narrow tetraquark states. It should be stressed
that two-body KK decays from any JP = 1+ uds̄s̄ state are forbidden because of
the conservation of the total spin and parity. The lowest threshold energy of the
allowed two-body decays is 1.39 GeV for the KK∗(895) channel. If the mass of the
JP = 1+ uds̄s̄ state lies below(closely to) the KK∗, two-meson decay channels are
(almost) closed, and hence, its width must be narrow.
The tetraquark uds̄s̄ states were discussed in Ref.[1], and noted as E(KK)-
mesons. In the MIT bag model,1) the theoretical mass for the isoscalar uds̄s̄(JP =
1+) state was predicted to be 1.65 GeV. Recently the isoscalar uds̄s̄ state with
JP = 1− was suggested in analogy with the Θ+ by Burns et al.,20) and its mass was
predicted to be ∼ 1.6 GeV. The isoscalar tetraquark uds̄s̄ in the flavor 1̄0 group was
called ϑ+-meson in Ref.[20] in the association with the Θ baryon. In this paper, we
investigate the ϑ+-meson(JP = 1+) with a constituent quark model. The theoretical
method of the calculations is the same as that applied to the study of pentaquark
and tetraquark in Refs.[17,21]. Namely, we apply the flux-tube quark model with
antisymmetrized molecular dynamics(AMD)22) to 4q systems. Based on the picture
of a flux-tube model, we ignore the coupling between the configurations (qq)3̄(q̄q̄)3
and (qq̄)1(qq̄)1, and solve the 4q dynamics with the variational method only in the
model space within the exotic color configuration (qq)3̄(q̄q̄)3. By assuming that the
f1 meson in the 1.4∼1.5 GeV mass region as the 4q state, we predict the mass and
the width of the tetraquark ϑ+(JP = 1+).
The method of AMD is a variational method. The adopted Hamiltonian is the
same as that of previous works.17), 21) The Coulomb and color-magnetic terms of the
OGE potential and the string potential are taken into account. The parameters in
the Hamiltonian are chosen to reasonably reproduce the normal hadron spectra.21)
In order to evaluate the ϑ+(1+) mass and the width, we adopt the observed data of
the f1-mesons(f1(1420) and f1(1510)) in the 1.4∼1.5 GeV region as an input. The
details of the formulations for fourquark systems are explained in Ref.[21].
As mentioned in,21) in order to predict the ϑ-meson mass, we need to deter-
mine the mass shift parameter M0 in the string potential for fourquark (qq)3̄(q̄q̄)3
systems. Here, we use the f1-meson mass as the input to determine the mass shift.
In the mass region 1∼1.6 GeV, three f1-mesons, f1(1285), f1(1420), and f1(1510)
are known, though the f1(1510) is not well established.
23) In the P -wave qq̄ state,
two f1-mesons are expected to appear in this energy region as the partners of the
qq̄ nonet. It is considered that the lower one is dominated by the light-quark uū
and dd̄ components(nn̄), and the major component of the higher one is the ss̄ state.
In the general interpretation, the lowest f1-meson(f1(1285)) is regarded as the nn̄
state. However, there are two candidates (f1(1420) and f1(1510)) for the qq̄ partner
of the f1(1285), and the assignment is not confirmed yet. In the constituent quark
model calculation,24) the masses of the two 1++ qq̄ states in the P -wave qq̄ nonet are
1.24 and 1.48 GeV. The theoretical mass spectra of the 1++ qq̄ states seems to be
consistent with the experimental ones if f1(1510) is assigned to be the partner of the
Letters 3
f1(1285) in the flavor nonet. This is consistent with the assignment in Ref.[25]. On
the other hand, an alternative interpretation of the f1(1285) and f1(1420) being the
qq̄ partners is claimed in Refs.[23,26,27]. These interpretations lead to an indication
that one of the f1(1420) or f1(1510) may be a non-qq̄ meson while the other and
f1(1285) can be understood as the partners of the conventional P -wave qq̄ states.
By ignoring the qq̄ annihilation, we calculate the mass of the JPC=1++ (us)3̄(ūs̄)3
state for the f1-meson with the present framework in the same way as for the
tetraquark ϑ-meson. We adjust the mass shift parameter M0 by fitting the mass
of the f1(1420) or the f1(1510). Then we get the ϑ
+(1+) mass around 1.4 GeV.
The obtained ϑ+(1+) wave function is dominated by the component with the
spin-zero (ud)3̄ and the spin-one (s̄s̄)3 in the spatially symmetric configuration, (0s)
This is consistent with the naive expectation in the diquark picture because the spin-
zero (ud)3̄ gain the color-magnetic interaction while only the spin-one configuration
is allowed in the spatially symmetric (s̄s̄)3 pair. As a result of the energy gain of the
color magnetic interaction, the ϑ+(1+) mass is slightly lower than the fourquark f1
mass.
Next we discuss the width of the ϑ+(1+) meson. As mentioned above, we suggest
that the ϑ+(1+)-meson may appear in the energy region ∼1.4 GeV near the KK∗
threshold. The expected decay modes are KK∗ and KKπ. If the branching into
KK∗ is small enough, the width should be narrow because the phase space for the
three-body decays is suppressed in general. In order to discuss the stability of the
ϑ+-meson, we consider only the two-hadron decay and give a rough estimation of
the ϑ+ width assuming that the coupling of the fourquark f1 with the KK̄
∗ and c.c.
is the same as that of the ϑ+ with the KK∗. By considering the phase space for the
two-body decays, we predict the width of ϑ+(1+) to be Γϑ = 20 ∼ 50 MeV.
In summary, we discussed the possibility of the JP = 1+ state of the isoscalar
tetraquark(S=+2), ϑ+-meson, with the uds̄s̄ content. If a f1 meson in the mass
region 1.4−1.5 GeV consists of four quarks nsn̄s̄, the mass of the isoscalar uds̄s̄(ϑ+-
meson) state with JP = 1+ is expected to be lower than that of the f1 meson. Within
a flux-tube quark model, a possible resonant state of uds̄s̄(JP = 1+) is suggested
to appear at ∼ 1.4 GeV with the width O(20 ∼ 50) MeV. We propose that the
ϑ+-meson is the good candidate for the tetraquark search, which would be observed
in the K+K+π− decay channel.
Recently, the θ+(1−) and the θ+(0+) were predicted by Burns et al.20) and by
Karlier and Lipkin,28) respectively. It should be pointed out that the allowed decay
channels are different among these three predictions JP = 1+, 1−, and 0+ states of
the ϑ+-mesons. The K+K+π− decay from the ϑ(JP = 1+) predicted in the present
work is suitable for the experimental tetraquark search.
As for the other candidates of the 4q states, it has been theoretically suggested
that the scalar mesons like f0(600), f0(980) and a0(980) below 1 GeV might be
interpreted as 4q states. We calculated the corresponding fourquarks with the present
framework. Then, we found that the masses of fourquarks with the exotic color
configurations (qq)3̄(q̄q̄)3 are much higher than these light scalar mesons. It indicates
that these scalar mesons might be other than the dominant (qq)3̄(q̄q̄)3 state, but
might be the hybrids of P -wave qq̄ and fourquark components with meson-meson
4 Letters
tails in the outer region as argued in Ref.[27].
The authors would like to thank to H. Nemura and H. Hidaka for the valuable
discussions. This work was supported by Japan Society for the Promotion of Science
and Grants-in-Aid for Scientific Research of the Japan Ministry of Education, Science
Sports, Culture, and Technology. The calculations of this work is supported by the
computer system in YITP.
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http://arxiv.org/abs/hep-ph/0404144
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