Decay of the Θ+ in a quark model

A. Hosakaa∗

aResearch Center for Nuclear Physics (RCNP), Osaka University, Ibaraki, Osaka567-0047, Japan

We study the decay of the pentaquark baryon in a non-relativistic quark model for po-sitive and negative parity Θ+ states. The decay amplitudes are expressed as products of aspectroscopic factor and an interaction matrix element. It is shown that relatively narrowwidths can be obtained for positive parity states, while the negative parity state of (0s)5

configuration couples strongly to the KN continuum states leading to very broad widths.It is also pointed out that the width of a 3/2− state should be strongly suppressed.

1. INTRODUCTION

The observation of an evidence for the pentaquark particle Θ+ by the LEPS group atSPring-8 has triggered an enormous amount of work both in experiments and theories [1].Investigations with new insights into the non-perturbative QCD dynamics are requiredfor the description of the exotic particles [2]. Although the experimental situation forthe existence of the pentaquarks requires further confirmation, some of the expectedproperties of the pentaquark baryons are very interesting to explain from the theoreticalpoint of view. Among them, the mechanism leading to the narrow width of the Θ+ isobviously an important issue to be understood together with the determination of its spinand parity [3].

In this report, as a physical quantity which reflects very sensitively the spin and parityand hence the internal structure of the Θ+ [4], we investigate the strong decay of theΘ+ [5]. To the lowest order, a fall-apart mechanism dominates where a rearrangement ofthe five quarks takes place: q4q̄(Θ+) → qq̄(K)q3(N).

Before going into details, we discuss briefly an important role of the centrifugal barrierfor the decaying KN state. For instance, if the Θ+ has spin-parity JP = 1/2±, the partialwave of the KN state is p or s wave depending on the parity. The different partial wavenature appears in the formula of the decay width

Γ+ =g2

KNΘ

2π

MNq3

EN(EN + MN)MΘ

, Γ− =(EN + MN)2

q2Γ+ , (1)

where Γ± denote the decay widths of the positive and negative parity Θ+, and gKNΘ

the KNΘ coupling constant. The factor in the second equation is about 50 for theΘ+(1540), and has a significant effect on the resulting decay width. For a typical strengthof gKNΘ ∼ 4 [5], we find Γ+ ∼ 20 MeV, while Γ− ∼ 1 GeV. This observation tells us that

∗E-mail: hosaka@rcnp.osaka-u.ac.jp

Nuclear Physics A 755 (2005) 407c–410c

0375-9474/$ – see front matter © 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.nuclphysa.2005.03.046

Lmqq

~N Θ

K K

N Θ

Figure 1. Decay of the pentaquark state.

if the spin-parity of the Θ+ is 1/2+, the narrow decay width may be explained rathernaturally, but if it is 1/2−, it would be rather difficult. In the following, we will verifythat the above naive observation indeed agrees with the quark model description.

2. DECAY WIDTH

The decay of the pentaquark state going into one baryon and one meson is dominatedby the fall-apart process shown in Figure 1 (left), which includes the annihilation of aquark-antiquark pair (right). The matrix element of such a process is written as a productof the so-called spectroscopic factor and an interaction matrix element,

MΘ+→KN = SKN · hint . (2)

The former term, SKN in Θ+ , is a probability amplitude to find in the pentaquark statethree-quark and quark-antiquark clusters having the quantum numbers of the nucleonand the kaon respectively,

|Θ+〉 = SKN |(qq̄)N(qqq)N〉 + · · · . (3)

Calculations obtaining this factor have been performed previously by several authors forsome specific configurations [6–8]. We summarize them in Table 1.

In the quark model, the interaction matrix element can be computed using the meson-quark interaction of the Yukawa type:

Lmqq = gq̄γ5λaφaq , (4)

where λa are SU(3) flavour matrices and φa are the octet meson fields. The coupling cons-tant g may be determined from the pion-nucleon coupling constant gπNN = 5g. Therefore,using gπNN ∼ 13, we find g ∼ 2.6. In this estimate, the meson-quark interaction of (4)

Table 1Spectroscopic factors SNK = 〈N(123)K(45̄)|Θ+〉 for various Θ+ configurations. The sumover the two isospin states, pK0 and nK+, is taken into account. For the notations of theconfigurations, see the caption of Table 2.

Configurations 1/2− 1/2+(SF) 1/2+(SC) 1/2+(JW)

SKN 1/2√

5/96√

5/192√

5/576

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Table 2Decay widths of the Θ+ in MeV for JP = 1/2±. In the columns denoted SF, SC

and JW, the results are presented for the configuration which minimizes the spin-flavourinteraction, which minimizes the spin-colour interaction and the configuration with theS = I = 0 diquark correlation of the Jaffe-Wilczek type. In this Table, numerical errorsin Ref. [5] which are not essential for the present discussion are corrected.

〈r2〉1/2 α20 1/2− 1/2+ (SF) 1/2+ (SC) 1/2+ (JW)

1/√

2 fm 6 fm−2 1500 41 21 7

1 fm 3 fm−2 890 63 32 11

is space-like, quark → quark and meson. On the contrary, the decay of the Θ+ needsan annihilation of a quark and an antiquark in the time-like region (right side of Figure1). Different kinematics may change the strength of the coupling because the meson is acomposite object. In the present estimate, we adopt however the value indicated above.

Further details on the calculation can be found in Ref. [5], and here several results aresummarized as follows. For the negative parity state of (0s)5, the decay width turns outto be of the order of several hundred MeV or more, typically ∼ 1 GeV, as shown in Table2. In the calculation it has been assumed that the spatial wave function for the initialand final state hadrons are described by the same harmonic oscillator states. Also themasses of the particles are taken to be their experimental values, e.g., MΘ+ = 1540 MeV.The size of the baryons 〈r2〉 is determined by the oscillator parameter α0. The very broadwidth suggests that the (0s)5 state couples very strongly to the KN continuum and ishardly identified with a resonant state with a narrow width.

For the positive parity states, the orbital excitation introduces more degrees of freedom,and four independent configurations are possible for spin-parity JP = 1/2+ [7]. Here weconsider three configurations: (SF) minimizing the spin-flavour interaction of one-mesonexchange [8], (SC) minimizing the spin-colour interaction of one-gluon exchange, and(JW) with a strong correlation in the S = I = 0 diquark channel as proposed by Jaffeand Wilczek [6]. As shown in Table 2, the resulting decay widths are about 80 MeV, 40MeV and 10 MeV, respectively. The diquark correlation of (JW) develops a spin-flavour-colour wave function having a small overlap with the decaying channel of the nucleonand the kaon. In the evaluation of these values, we did not consider spatial correlations.However, if, for instance, a diquark correlation is developed, the spatial overlap becomesless than unity, further suppressing the decay width.

3. SUMMARY AND DISCUSSIONS

In this report we have discussed the decay of the Θ+ pentaquark. The mechanism ofthe fall-apart process is very sensitive to the structure. For JP = 1/2−, the naive groundstate of (0s)5 can no longer survive as a narrow resonance as it couples very strongly withthe KN scattering state. In other words, the (0s)5 configuration is almost the scatteringstate with little component of a confined state [9]. On the contrary, the decay widths

A. Hosaka / Nuclear Physics A 755 (2005) 407c–410c 409c

of 1/2+ states turn out to be of the order of ten MeV, but with once again a strongdependence on the configuration.

The present analysis can be extended straightforwardly to the Θ+ of spin 3/2. For thenegative parity state, the spin 1 state of the four quarks in the Θ+ may be combinedwith the spin of the s̄ to form the total spin 3/2. In this case the final KN state mustbe in a d-wave state, and therefore the spectroscopic factor for finding a d-wave KNstate in the (0s)5 is simply zero. If a tensor interaction induces an admixture of a d-waveconfiguration, it can decay into a d-wave KN state. However, the mixture of the d-wavestate is expected to be small, just as for the deuteron. There could be a possible decaychannel to the nucleon and the vector K∗ of JP = 1−. This decay, however, does notoccur since the total mass of the decay channel is larger than the mass of the Θ+. Hencethe JP = 3/2− state could be another candidate for the observed narrow state. This statedoes not have a spin-orbit partner and forms a single resonance peak around its energy.For the positive parity case, the p-state orbital excitation may be combined with the spinof the s̄ to form the total spin 3/2. In this case, the calculation of the decay width isprecisely the same as before (see Ref. [5] for more details). After taking the average overthe angle �q, however, the coupling yields the same factor as for the case J = 1/2. Hencethe decay rate of the spin 3/2 Θ+ is the same as that of Θ+ of spin 1/2 in the presenttreatment.

AcknowledgementsThe author would like to thank E. Hiyama, T. Hyodo, M. Kamimura, T. Nakano,

S.I. Nam, M. Oka, T. Shinozaki and H. Toki, for stimulating discussions. This workwas supported in part by the Grant for Scientific Research ((C) No.16540252) from theMinistry of Education, Culture, Science and Technology, Japan.

REFERENCES

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2. M. Oka, Prog. Theor. Phys. 112 (2004) 1 and references therein.3. A. W. Thomas, K. Hicks and A. Hosaka, Prog. Theor. Phys. 111 (2004) 291.4. A. Hosaka, Phys. Lett. B 571 (2003) 55.5. A. Hosaka, M. Oka and T. Shinozaki, arXiv:hep-ph/0409102.6. R. L. Jaffe and F. Wilczek, Phys. Rev. Lett. 91 (2003) 232003.7. B.K. Jennings and K. Maltman, Phys. Rev. D 68 (2004) 094020.8. C. E. Carlson, C. D. Carone, H. J. Kwee and V. Nazaryan, Phys. Rev. D 70 (2004)

037501.9. E. Hiyama, talks given at the international workshop PENTAQUARK04, June 20-

23 (2004) Spring-8 and at the international conference BARYONS04, October 25-29(2004) Paris and to be published in the proceedings.

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