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A diquark model for pentaquark baryons and tetraquark mesons
Physics Letters B 600 (2004) 215–222 www.elsevier.com/locate/physletb
A diquark model for pentaquark baryons and tetraquark mesons R.S. Kaushal a,c , D. Parashar b,c , A.K. Sisodiya c a Department of Physics, Ramjas College (University Enclave), University of Delhi, Delhi-110007, India b Department of Physics, A.R.S.D. College, University of Delhi, New Delhi-110021, India c Department of Physics and Astrophysics, University of Delhi, Delhi-110007, India
Received 29 June 2004; received in revised form 27 August 2004; accepted 9 September 2004 Available online 15 September 2004 Editor: P.V. Landshoff
Abstract In the spirit of quark–diquark model, used earlier for the nucleon [Ann. Phys. (N.Y.) 108 (1977) 198] we investigate here some properties of the newly discovered exotic pentaquark systems, viz., θ + (1540) and (say) ζ 0 (3099). For this purpose, in the first stage, we consider these multiquark states as three-body systems—two diquarks of (ud)-type and an antiquark with a Coulomb-type color force between them. In the second stage, we replace this three-body system with an effective two-body system in which the two bound diquarks constitute the core and the extra (fifth) antiquark (valence) revolves around this core in a Coulomb field. After accounting for the extended character of the diquark from the earlier work, we compute the form factors for these pentaquark systems and make some remarks about their magnetic moments. Predictions are also made for the form factors of the four-quark (tetraquark) mesons, i.e., for the diquark–antidiquark bound systems. In both θ + (1540) and ζ 0 (3099) the valance antiquarks are found to be embedded in the core unlike the case of nucleon. 2004 Elsevier B.V. All rights reserved. PACS: 12.39.Mk; 13.40.Gp; 13.40.Em Keywords: Pentaquark systems; Quark–double-diquark model; Charge distribution; Form factor
1. Introduction In physics sometimes even oversimplified models can provide reasonably good theoretical results which, in turn, are found to be in tune with exper-
E-mail address: [email protected] (R.S. Kaushal). 0370-2693/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2004.09.014
iments. While corrections to these models further improve the agreement with experiments, the lowestorder results are suggestive enough to understand the underlying mechanism of the system. One such model, proposed [1] some 25 years ago for the nucleon, is the quark–diquark (QDQ) model in which the three-quark system is replaced with an effective two-body system—one diquark constituting the core
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and the third quark (moving around the core) acting as the valence quark. Further, only the Coulomb part of the quark confining two-step potential [2–5], viz., V (r) = −β/r,
r B,
= −V0 + kr,
r > B,
(1a) (1b)
between the quarks is found sufficient to explain, at least qualitatively, the various ground state properties not only of proton and neutron but also that of deuteron. In fact, the same potential has been successfully used for the q q¯ pair to study the properties of pion [2], kaon [3] and rather heavier (charmed) mesons [4,5]. In exactly the same spirit now we shall employ this potential to the study of the pentaquark and the four-quark (the so-called tetraquark) systems like the diquark–antidiquark bound states, which are desperately waiting to be discovered. The main ingredients entering the QDQ model as inputs used at that time embody the dipole behaviour of the proton form factor, namely, p GE Q2 =
1 (1 + Q2 /18.1)2
(Q2 in fm−2 ), and the known negative mean square charge radius, η, of the neutron, 0.11 < η < 0.16 (in fm2 ). To the best of our knowledge these experimental features of the nucleon form factors are still valid even today. As far as the role of the linear part in potential (1) is concerned, the same had been investigated [5] subsequently in relation to other properties of baryons in general. In the earlier work [1] (referred to as KK) on nucleon, a common core of (ud) diquark was considered and the third quark (u in the proton and d in the neutron) was supposed to revolve around this core. A Coulomb-type (color) potential (1a) between the quarks in the core as also between the core and the valence is considered to be operating, which eventually leads to the diquark charge density in the ground state as 1 2r exp − , ρD (r) = (2a) 3 πboc boc or the corresponding charge form factor as
1 FD,1 Q2 = , 1 2 2 2 1 + 4 Q boc
(2b)
where boc = 0.47 fm and Q2 is the momentum— transferred square. These structural forms are found to be consistent with experiments as far as the explanation of the ground state properties of proton, neutron, and deuteron is concerned for an effective quark mass mq = 513 MeV. Further, it is found that the valence u quark in proton is embedded in the core (corresponding to boc = bov = 0.47 fm) and the valence d quark in neutron lies outside the core. (This point is clear from the two sets of boc and bov for neutron in KK and also from the modified calculation in which the same core size (boc = 0.47 fm) for both proton and neutron is retained, furnishing the value bov = 0.577 fm for neutron.) Thus the diquark mass, MD = 2mq − |E1c |, where |E1c |, is the magnitude of diquark binding energy, turns out to be 681.3 MeV. In the present work we extend this scheme of the QDQ model of nucleon to the case of the newly discovered [6–9] exotic pentaquark systems. One such system, confirmed in different experiments [6], is denoted by θ + and identified as a (uudd s¯) composite with a mass of 1540 MeV, whereas the other is expected [7] to be a (uudd c) ¯ formation with a mass of 3099 MeV. We designate this latter particle as ζ 0 . Yet another one waiting confirmation is the heavy Ξ version with a charge of −2 at about 1860 MeV [8]. For the present, however, we confine ourselves only to the study of θ + and ζ 0 . With regard to the production [9] and properties of θ + , several studies [10–15] have been carried out recently. The problem pertaining to the charge distribution and from factor of these newly discovered particles have, however, not yet been addressed. In Section 2, we discuss a quark double-diquark (QDDQ) model for these pentaquark baryons. After using the knowledge of diquark from KK, we make possible predictions for the charge distribution, charge form factor, and the magnetic moment of these pentaquark systems. For the sake of completeness, some predictions are also made in Section 3 for the tetraquark systems within this framework, namely, the four-quark (bound state) mesons. Some remarks on the multiquark configuration in pentaquark systems are made in Section 4. Finally, we discuss our results in Section 5.
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2. Quark–double-diquark model of pentaquark systems In analogy with the QDQ model of nucleon, we propose here a quark double-diquark (QDDQ) model for pentaquark systems. According to this model these systems are now considered as three-body systems— two diquarks and one antiquark. Not only this, as for the nucleon, this three-body system is again replaced with an effective two-body system—a core consisting of two (identical) bound diquarks and a valence antiquark revolving around this core. This valence quark is s¯ in θ + and c¯ in ζ 0 . Analogous to the nucleon core, we consider here Coulomb-type color force described by V (r) = −δ/r, operating between the diquarks and between the core and the valence with respect to the common center-of-mass of the diquarks. Corresponding to the core and the valence structures, different couplings like δc and δv (or different Bohr radii Boc and Bov ) will appear in the hydrogenic wavefunctions. Interestingly, the importance of a similar diquark model for the pentaquark systems has recently been emphasized by Jaffe and Wilczek [14] and in a certain form also by Karliner and Lipkin [10]. Following the general prescriptions of KK, we write down the charge distribution for θ + and ζ 0 systems as 2 1 ρθ (r ) = ρc (r ) + ρv (r ), (3) 3 3 2 2 ρζ (r ) = ρc (r ) − ρv (r ), (4) 3 3 and the corresponding charge form factors as 1 1 2FD,1 (Q2 ) p 2 GE Q = + 2 , 3 1 + 1 Q2 B 2 2 1 + 1 Q2 B 2 4
oc
4
oc
(5)
2 FD,1 1 ζ GE Q2 = − . 1 2 2 2 3 1 + 1 Q2 B 2 2 1 + 4 Q Boc oc 4 (6) Here we have appropriately accounted for the extended character of the diquark by writing the core part of the charge distributions in (3) and (4) as ρc (r ) = ρD (r )ρ1 (r − r ) d r (7) (Q2 )
and using the folding-theorem of Fourier transforms in computing the form factors (5) and (6). Note that the
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form of the charge density is basically derived from the radial part of the hydrogenic wave functions, viz., 1 2r exp − ρ1 (r ) = B0 πB03 and each ρ in (7) is normalized separately to unity. Other relations which we frequently use are the definition of first Bohr radii Boc and Bov as Boc = 1/µc δc and Bov = 1/µv δv , where µc and µv , the reduced masses for the core and the core-valence systems, respectively, are given by µc = (MD /2), and µv =
2MD ms¯ 2MD + ms¯
for θ +
2MD mc¯ 2MD + mc¯
for ζ 0 .
and µv =
Within this framework, we obtain the masses of θ + and ζ 0 systems through the relations Mθ = 2MD − E1c + ms¯ − E1v , (8a) c v Mζ = 2MD − E1 + mc¯ − E1 , (8b) where |E1c | and |E1v | are the ground state energies of the core and the core-valence systems and are given by |E1c | = 12 µc δc2 , |E1v | = 12 µv δv2 and |E1v | = 12 µv δv 2 . Since the valences in θ + and ζ 0 are considered to be different, the primed notations stand for ζ 0 . In θ + and ζ 0 systems, while we treat the cores to be identical, the valances are, however, considered different unlike the case of QDQ model of nucleon. In all there are three but inter-related unknown parameters in QDDQ model, namely, the first Bohr radii Boc , , or, equivalently the corresponding couplings Bov , Bov δc , δv , δv . Two of these parameters, of course, can be fixed from (8a) and (8b) using experimental values for the masses 1540 MeV and 3099 MeV for Mθ and Mζ , and standard values [16] of ms¯ and mc¯ as 500 MeV and 1600 MeV, respectively. For the computation of the form factors and the corresponding charge distributions from (3)–(6), we follow the procedure delineated hereunder. In case of θ + (1540) Eq. (8a) is used to determine δv for some representative values of δc such as 0.723 (or Boc = 0.8 fm) and the known values of Mθ , MD and ms¯ . The value of δv (or that of Bov = 0.48 fm) turns out to be 1.13. The predictions for the charge distribution and form factor of θ + are shown in Figs. 1 and 2,
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Fig. 3. Charge distribution of ζ 0 (3099) with constituent diquarks in the excited state. Fig. 1. Charge distribution of θ + (1540) in the QDDQ model.
Fig. 2. Form factor of θ + (1540) in the QDDQ model.
respectively. Similarly, if we substitute the values of Mζ , MD , mc¯ and the above values of δc , the value of δv from Eq. (8b) turns out to be imaginary—implying the unstable nature of the corresponding system (i.e., ζ 0 ). In other words, the core with two diquarks in the ground state is unable to hold the heavy mass valence of anticharm quark of mass 1600 MeV. This is, however, possible if the core diquarks are in the (at least first) excited state. In that case diquark becomes more massive, MD∗ = 2mq − |E2c |, corresponding to the 2sstate. Thus, MD∗ in the QDQ model turns out to be MD∗ = 939.8 MeV and Eq. (8b) now reads as Mζ = 2MD∗ − E1c + mc¯ − E1v . (9) (or δ ) for the given B From here the values of Bov oc v (= 0.8 fm) work out to be 0.27 fm (δc = 0.86). This is indeed much smaller than Boc which in turn implies that the c¯ quark is sitting near the center. This fact is also reflected in the charge distribution of ζ 0 (cf. Fig. 3). Thus for ζ 0 there appears to be a large
Fig. 4. Form factor of ζ 0 (3099) with constituent diquarks in the excited state.
negative charge in the central region and the positive charge remains in the outer periphery. Note that corresponding to the 2s-diquark, the diquark form factor in the first term of (6) should now be read as 2 )(1 − 2Q2 b 2 ) (1 − Q2 boc oc , FD,2 Q2 = 2 )4 (1 + Q2 boc
(10)
which actually corresponds to the diquark charge distribution in the 2s-state, viz., 1 r 2 exp(−r/bo ). 2 − ρD,2 (r) = (11) 32πbo3 bo The form factor of ζ 0 (3099) obtained from (6) is displayed in Fig. 4. For the sake of curiosity, we also demonstrate the charge distribution and the charge form factor of θ + (1540) corresponding to the 2sdiquark in the core in Figs. 5 and 6, respectively. Finally, we wish to append some remarks on the magnetic moments of these pentaquark baryons in this oversimplified QDDQ model. If we assume that the contribution to the magnetic moment of these pen-
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219
3. Form factor of the tetraquark systems
Fig. 5. Charge distribution of θ + (1540) with constituent diquarks in the excited state.
As concerted efforts are already going on currently at the experimental level to look for possible fourquark bound states, we are tempted to extend the above results to the case of these tetraqurak mesons. We consider these tetraquark mesons as bound states of a di¯ that in fact furnishes quark (D) and an antidiquark (D) a color singlet state, in consonance with our expectation. It is now a matter of straightforward extension of the above results to the case of the diquark (ud) and ¯ constituting the new exotic mesons. antidiquark (u¯ d) In fact the form factor for these exotic mesons, after using the results of KK for the diquark, is given by ¯ 2 D GD E Q =
FD,1 (Q2 ) 1 + 14 Q2 BoDD¯
2
where FD,1 (Q2 ) is given by (2b). Naturally, the cou¯ pling (and hence the Bohr radius BoDD¯ ) for the DD system here is expected to be different from that for the DD system of QDDQ model. The results are presented for the charge distribution in Fig. 7 and the charge form factor in Fig. 8. Fig. 6. Form factor of θ + (1540) with constituent diquarks in the excited state.
4. Multiquark configurations in pentaquark systems taquark baryons comes mainly from the spin-1/2 constituents, then it is the valence quark that will con¯ Thus tribute to it. In θ + it is s¯ whereas in ζ 0 it is c. we expect [17] the magnetic moments to be given as µθ + = +(Mθ /ms¯ ) nm ∼ = 3.12 nm, µζ 0 = −(Mζ /mc¯ ) nm ∼ = −1.94 nm. In the absence of any experimental information on magnetic moments of these baryons it is difficult to draw any definite conclusions regarding the relative magnitudes of the normal and anomalous contributions to the above values of µθ and µζ . It is equally difficult to estimate the role played by the core and the valence in these numerical values. It is, however, abundantly clear that the anomalous part is going to be quite significant for these systems.
With a view to describe various properties of 3quark baryons several models with varying quark configurations (such as sea-valence model, QDQ model, bag or cloudy bag model, etc.) have been used in the past. The problem is, however, more complicated with pentaquark systems. While more experimental results in near future on these systems will act as the guiding principle in this regard, the modest attempt made here in terms of QDDQ model is based on the work of KK [1] and it also embodies the spirit of the recent suggestions of Jaffe and Wilczek [14]. Baryon resonances with strangeness +1 (denoted by Z ∗ ) have also been studied by Jennings and Maltman [18]. They have explored the implications of a number of models compatible with the Z ∗ resonances in the region of θ + (1540) for the as-yet-undetermined quantum numbers of θ + . According to them θ + is the lowest resonance in this region with a width [19] Γ < 6 MeV.
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Fig. 7. Charge distribution of the diquark–antidiquark mesons (a case of tetraquark systems).
boson exchange between the constituent quarks and (ii) when the qq interactions are generated by an effective color-magnetic exchange. No doubt, the predictions are made corresponding to both the “schematic” and the “realistic” situations with regard to the spacedependence of the interactions, but an account of the two-body energies (the ones attributed to the diquark– diquark interactions) is not quite transparent in these calculations. On the other hand, if diquark configurations have to play certain important role (and they have to, as they are also the colored objects) then an account of these two (and perhaps higher)—body energies is necessary. In what follows we shall briefly outline our recent results [20] obtained schematically for these energies in the context of diquark clustering in quark–gluon plasma [21]. While details will appear elsewhere, this is mainly to get a glimpse in to the relative contributions arising from different spin-color diquark states. Evidently, this offers a clue towards a better comprehension of the underlying mechanism in these pentaquark systems. As a matter of fact, starting with the following type of color hyperfine interaction Hamiltonian i · λ j σi · σj , λ Hint = −A i,j
Fig. 8. Form factor of the diquark–antidiquark mesons (a case of tetraquark systems).
While a unified description for both θ + and ζ 0 is suggested in this Letter, a link of QDDQ configuration with the diquark–triquark configuration of θ + proposed by Karliner and Lipkin [10] can easily be established. A straightforward spin-analysis of these pentaquark baryons reveals that the double-diquark system spends 50% of its time in spin-1 configuration. The physical mechanism responsible for this possibility could be that in the vicinity of an antiquark (¯s in θ + and c¯ in ζ 0 ) a spin-0 (ud) diquark gets polarized, mainly because of the increase in the color hyperfine binding energy, presumably as a result of spin-flip. This fact is further corroborated by our recent studies [18] on the diquark clustering in quark– gluon plasma. For the pentaquark systems Jennings and Maltman [18] use the “hyperfine” interactions and account for two cases, namely, (i) when the spin-dependent qq interactions are generated by an effective Goldstone
where A is the strength of interaction and σ ’s and λ’s are the usual Pauli and Gell-Mann matrices, respectively, the interaction energies are computed for all permissible spin-color states of a two-diquark system in the cluster of quarks and in the light of a generalized Pauli Principle. The results are summarized in Table 1 for various spin-color states of two diquarks. Notice that the calculations yield the interaction energy as (14A/3) for the spin-flip case which is larger than the maximum value (4A) obtained for the nonspin-flip case. This is true especially for color states
, |3a ≡ , |15sa ≡ |3s ≡ . However, for other color states, namely,
|6as ≡ , , |15s ≡
, |15as ≡ , |6s ≡
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Table 1 Two-body (diquark–diquark) interaction energies in multiquark systems in different spin–color states Spin states Color states
|1a
|5
|3s
|1s
|3a
|3a , |15sa , |3s
3A
4A
2A
A
14 A 3
|6as , |6s , |15as
5A 2
2A
13 A 3 20 A 3
37 A 6 34 A 3
3A
|15s
2A 3 − 38 A
4A 3
the situation is found to be different. Thus, the former case of spin-flip seems to favour the physical mechanism mentioned above.
221
In the absence of any concrete experimental information available on the properties of θ + and ζ 0 particles studied in this Letter, no meaningful comparison is therefore possible for these predictions. However, the results presented in this communication are at best at the level of speculation, nevertheless they can serve as a useful guide for the future when experimental situation becomes free from ambiguities. A study of the QDDQ model for Ξ −− (1860) particle in terms of its (dsds u) ¯ constituents is in progress.
Acknowledgements 5. Concluding discussion In analogy with the QDQ model for nucleons, we have ventured in this Letter to look into the structural details of the newly discovered exotic pentaquark baryons by way of proposing a QDDQ model for these systems. Extended character of the diquark is appropriately accounted for in predicting the charge distribution and charge form factor for θ + (1540) and ζ 0 (3099) particles. While the results obtained are consistent for the θ + baryon, some abnormal behavior is observed for the ζ 0 baryon in the sense that the heavy valence c¯ in this case is found to remain well within the core and is, therefore, capable to interact only with the excited diquarks in the core. As a matter of fact similar speculations have been made by Jaffe and Wilczek [14]. Within this framework, the magnetic moments for θ + and ζ 0 are predicted to be +3.12 nm and −1.94 nm, respectively. In fact an attempt has been made in the preceding section to find a plausible basis for these double diquark configurations, which in turn couple with the remaining (fifth) antiquark. Based on the situation analogous to other mesons, which are basically the bound states of a quark and an antiquark, we have sought here to expand the ambit to incorporate the possibility of studying the multiquark (four-quark or tetraquark) mesons as bound state of a diquark and an antidiquark. The predictions are made for the charge distribution and the form factor for the ¯ system within this scheme. Of course, bound (ud)(u¯ d) the extended character of the diquark has also been included in these calculations.
Authors wish to thank Professor V.S. Bhasin for several useful discussions. One of us (A.K.S.) gratefully acknowledges the Senior Research Fellowship of CSIR, New Delhi.
References [1] R.S. Kaushal, D.S. Kulshreshtha, Ann. Phys. (N.Y.) 108 (1977) 198. [2] R.S. Kaushal, Phys. Lett. B 57 (1975) 354. [3] D.S. Kulshreshtha, R.S. Kaushal, Phys. Lett. B 102 (1981) 185. [4] R.S. Kaushal, Phys. Lett. B 60 (1975) 81; R.S. Kaushal, D.S. Kulshreshtha, D. Parashar, Nuovo Cimento A 37 (1977) 55; R.S. Kaushal, D.S. Kulshreshtha, Nuovo Cimento A 40 (1977) 163. [5] D.S. Kulshreshtha, R.S. Kaushal, Phys. Rev. D 26 (1982) 2331. [6] T. Nakano, et al., LEPS Collaboration, Phys. Rev. Lett. 91 (2003) 012002; Also see: B. Schwarzschild, Phys. Today 2 (2004) 19. [7] A. Aktas, et al., H1 Collaboration, hep-ex/0403017; Also see: Phys. World 17 (2004) 3. [8] C. Alt, et al., NA49 Collaboration, Phys. Rev. Lett. 92 (2004) 042003, hep-ex/0310014. [9] V. Kubarovsky, et al., CLAS Collaboration, Phys. Rev. Lett. 92 (2004) 032001. [10] Y. Oh, H. Kim, S.H. Lee, Phys. Rev. D 69 (2004) 014009; Q. Zhao, Phys. Rev. D 69 (2004) 053009; M. Karliner, H.J. Lipkin, Phys. Lett. B 575 (2003) 249; M. Karliner, H.J. Lipkin, hep-ph/0307243. [11] R.N. Cahn, G.H. Trilling, Phys. Rev. D 69 (2004) 011501 (R). [12] P. Bicudo, G.M. Marques, Phys. Rev. D 69 (2004) 011503 (R). [13] A.W. Thomas, K. Hicks, A. Hosaka, Prog. Theor. Phys. 111 (2004) 291(L). [14] R.L. Jaffe, F. Wilczek, Phys. Rev. Lett. 91 (2003) 232003; Also see: R.L. Jaffe, F. Wilczek, Phys. World 6 (2004). [15] P.-Z. Huang, et al., hep-ph/0401191.
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[16] D.H. Perkins, Introduction to High Energy Physics, fourth ed., Oxford Univ. Press, Oxford, 2000, p. 24. [17] E.M. Henley, W. Thirring, Elementary Quantum Field Theory, McGraw–Hill, New York, 1962, p. 230. [18] B.K. Jennings, K. Maltman, Phys. Rev. D 69 (2004) 094020. [19] S. Nussinov, hep-ph/0307357.
[20] A.K. Sisodiya, V.S. Bhasin, R.S. Kaushal, On diquark clustering in quark–gluon plasma, in preparation. [21] J.F. Donoghue, K.S. Sateesh, Phys. Rev. D 38 (1988) 360; S.K. Karn, R.S. Kaushal, Y.K. Mathur, Eur. Phys. J. C 14 (2000) 487, and references therein.
A diquark model for pentaquark baryons and tetraquark mesons R.S. Kaushal a,c , D. Parashar b,c , A.K. Sisodiya c a Department of Physics, Ramjas College (University Enclave), University of Delhi, Delhi-110007, India b Department of Physics, A.R.S.D. College, University of Delhi, New Delhi-110021, India c Department of Physics and Astrophysics, University of Delhi, Delhi-110007, India
Received 29 June 2004; received in revised form 27 August 2004; accepted 9 September 2004 Available online 15 September 2004 Editor: P.V. Landshoff
Abstract In the spirit of quark–diquark model, used earlier for the nucleon [Ann. Phys. (N.Y.) 108 (1977) 198] we investigate here some properties of the newly discovered exotic pentaquark systems, viz., θ + (1540) and (say) ζ 0 (3099). For this purpose, in the first stage, we consider these multiquark states as three-body systems—two diquarks of (ud)-type and an antiquark with a Coulomb-type color force between them. In the second stage, we replace this three-body system with an effective two-body system in which the two bound diquarks constitute the core and the extra (fifth) antiquark (valence) revolves around this core in a Coulomb field. After accounting for the extended character of the diquark from the earlier work, we compute the form factors for these pentaquark systems and make some remarks about their magnetic moments. Predictions are also made for the form factors of the four-quark (tetraquark) mesons, i.e., for the diquark–antidiquark bound systems. In both θ + (1540) and ζ 0 (3099) the valance antiquarks are found to be embedded in the core unlike the case of nucleon. 2004 Elsevier B.V. All rights reserved. PACS: 12.39.Mk; 13.40.Gp; 13.40.Em Keywords: Pentaquark systems; Quark–double-diquark model; Charge distribution; Form factor
1. Introduction In physics sometimes even oversimplified models can provide reasonably good theoretical results which, in turn, are found to be in tune with exper-
E-mail address: [email protected] (R.S. Kaushal). 0370-2693/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2004.09.014
iments. While corrections to these models further improve the agreement with experiments, the lowestorder results are suggestive enough to understand the underlying mechanism of the system. One such model, proposed [1] some 25 years ago for the nucleon, is the quark–diquark (QDQ) model in which the three-quark system is replaced with an effective two-body system—one diquark constituting the core
216
R.S. Kaushal et al. / Physics Letters B 600 (2004) 215–222
and the third quark (moving around the core) acting as the valence quark. Further, only the Coulomb part of the quark confining two-step potential [2–5], viz., V (r) = −β/r,
r B,
= −V0 + kr,
r > B,
(1a) (1b)
between the quarks is found sufficient to explain, at least qualitatively, the various ground state properties not only of proton and neutron but also that of deuteron. In fact, the same potential has been successfully used for the q q¯ pair to study the properties of pion [2], kaon [3] and rather heavier (charmed) mesons [4,5]. In exactly the same spirit now we shall employ this potential to the study of the pentaquark and the four-quark (the so-called tetraquark) systems like the diquark–antidiquark bound states, which are desperately waiting to be discovered. The main ingredients entering the QDQ model as inputs used at that time embody the dipole behaviour of the proton form factor, namely, p GE Q2 =
1 (1 + Q2 /18.1)2
(Q2 in fm−2 ), and the known negative mean square charge radius, η, of the neutron, 0.11 < η < 0.16 (in fm2 ). To the best of our knowledge these experimental features of the nucleon form factors are still valid even today. As far as the role of the linear part in potential (1) is concerned, the same had been investigated [5] subsequently in relation to other properties of baryons in general. In the earlier work [1] (referred to as KK) on nucleon, a common core of (ud) diquark was considered and the third quark (u in the proton and d in the neutron) was supposed to revolve around this core. A Coulomb-type (color) potential (1a) between the quarks in the core as also between the core and the valence is considered to be operating, which eventually leads to the diquark charge density in the ground state as 1 2r exp − , ρD (r) = (2a) 3 πboc boc or the corresponding charge form factor as
1 FD,1 Q2 = , 1 2 2 2 1 + 4 Q boc
(2b)
where boc = 0.47 fm and Q2 is the momentum— transferred square. These structural forms are found to be consistent with experiments as far as the explanation of the ground state properties of proton, neutron, and deuteron is concerned for an effective quark mass mq = 513 MeV. Further, it is found that the valence u quark in proton is embedded in the core (corresponding to boc = bov = 0.47 fm) and the valence d quark in neutron lies outside the core. (This point is clear from the two sets of boc and bov for neutron in KK and also from the modified calculation in which the same core size (boc = 0.47 fm) for both proton and neutron is retained, furnishing the value bov = 0.577 fm for neutron.) Thus the diquark mass, MD = 2mq − |E1c |, where |E1c |, is the magnitude of diquark binding energy, turns out to be 681.3 MeV. In the present work we extend this scheme of the QDQ model of nucleon to the case of the newly discovered [6–9] exotic pentaquark systems. One such system, confirmed in different experiments [6], is denoted by θ + and identified as a (uudd s¯) composite with a mass of 1540 MeV, whereas the other is expected [7] to be a (uudd c) ¯ formation with a mass of 3099 MeV. We designate this latter particle as ζ 0 . Yet another one waiting confirmation is the heavy Ξ version with a charge of −2 at about 1860 MeV [8]. For the present, however, we confine ourselves only to the study of θ + and ζ 0 . With regard to the production [9] and properties of θ + , several studies [10–15] have been carried out recently. The problem pertaining to the charge distribution and from factor of these newly discovered particles have, however, not yet been addressed. In Section 2, we discuss a quark double-diquark (QDDQ) model for these pentaquark baryons. After using the knowledge of diquark from KK, we make possible predictions for the charge distribution, charge form factor, and the magnetic moment of these pentaquark systems. For the sake of completeness, some predictions are also made in Section 3 for the tetraquark systems within this framework, namely, the four-quark (bound state) mesons. Some remarks on the multiquark configuration in pentaquark systems are made in Section 4. Finally, we discuss our results in Section 5.
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2. Quark–double-diquark model of pentaquark systems In analogy with the QDQ model of nucleon, we propose here a quark double-diquark (QDDQ) model for pentaquark systems. According to this model these systems are now considered as three-body systems— two diquarks and one antiquark. Not only this, as for the nucleon, this three-body system is again replaced with an effective two-body system—a core consisting of two (identical) bound diquarks and a valence antiquark revolving around this core. This valence quark is s¯ in θ + and c¯ in ζ 0 . Analogous to the nucleon core, we consider here Coulomb-type color force described by V (r) = −δ/r, operating between the diquarks and between the core and the valence with respect to the common center-of-mass of the diquarks. Corresponding to the core and the valence structures, different couplings like δc and δv (or different Bohr radii Boc and Bov ) will appear in the hydrogenic wavefunctions. Interestingly, the importance of a similar diquark model for the pentaquark systems has recently been emphasized by Jaffe and Wilczek [14] and in a certain form also by Karliner and Lipkin [10]. Following the general prescriptions of KK, we write down the charge distribution for θ + and ζ 0 systems as 2 1 ρθ (r ) = ρc (r ) + ρv (r ), (3) 3 3 2 2 ρζ (r ) = ρc (r ) − ρv (r ), (4) 3 3 and the corresponding charge form factors as 1 1 2FD,1 (Q2 ) p 2 GE Q = + 2 , 3 1 + 1 Q2 B 2 2 1 + 1 Q2 B 2 4
oc
4
oc
(5)
2 FD,1 1 ζ GE Q2 = − . 1 2 2 2 3 1 + 1 Q2 B 2 2 1 + 4 Q Boc oc 4 (6) Here we have appropriately accounted for the extended character of the diquark by writing the core part of the charge distributions in (3) and (4) as ρc (r ) = ρD (r )ρ1 (r − r ) d r (7) (Q2 )
and using the folding-theorem of Fourier transforms in computing the form factors (5) and (6). Note that the
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form of the charge density is basically derived from the radial part of the hydrogenic wave functions, viz., 1 2r exp − ρ1 (r ) = B0 πB03 and each ρ in (7) is normalized separately to unity. Other relations which we frequently use are the definition of first Bohr radii Boc and Bov as Boc = 1/µc δc and Bov = 1/µv δv , where µc and µv , the reduced masses for the core and the core-valence systems, respectively, are given by µc = (MD /2), and µv =
2MD ms¯ 2MD + ms¯
for θ +
2MD mc¯ 2MD + mc¯
for ζ 0 .
and µv =
Within this framework, we obtain the masses of θ + and ζ 0 systems through the relations Mθ = 2MD − E1c + ms¯ − E1v , (8a) c v Mζ = 2MD − E1 + mc¯ − E1 , (8b) where |E1c | and |E1v | are the ground state energies of the core and the core-valence systems and are given by |E1c | = 12 µc δc2 , |E1v | = 12 µv δv2 and |E1v | = 12 µv δv 2 . Since the valences in θ + and ζ 0 are considered to be different, the primed notations stand for ζ 0 . In θ + and ζ 0 systems, while we treat the cores to be identical, the valances are, however, considered different unlike the case of QDQ model of nucleon. In all there are three but inter-related unknown parameters in QDDQ model, namely, the first Bohr radii Boc , , or, equivalently the corresponding couplings Bov , Bov δc , δv , δv . Two of these parameters, of course, can be fixed from (8a) and (8b) using experimental values for the masses 1540 MeV and 3099 MeV for Mθ and Mζ , and standard values [16] of ms¯ and mc¯ as 500 MeV and 1600 MeV, respectively. For the computation of the form factors and the corresponding charge distributions from (3)–(6), we follow the procedure delineated hereunder. In case of θ + (1540) Eq. (8a) is used to determine δv for some representative values of δc such as 0.723 (or Boc = 0.8 fm) and the known values of Mθ , MD and ms¯ . The value of δv (or that of Bov = 0.48 fm) turns out to be 1.13. The predictions for the charge distribution and form factor of θ + are shown in Figs. 1 and 2,
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Fig. 3. Charge distribution of ζ 0 (3099) with constituent diquarks in the excited state. Fig. 1. Charge distribution of θ + (1540) in the QDDQ model.
Fig. 2. Form factor of θ + (1540) in the QDDQ model.
respectively. Similarly, if we substitute the values of Mζ , MD , mc¯ and the above values of δc , the value of δv from Eq. (8b) turns out to be imaginary—implying the unstable nature of the corresponding system (i.e., ζ 0 ). In other words, the core with two diquarks in the ground state is unable to hold the heavy mass valence of anticharm quark of mass 1600 MeV. This is, however, possible if the core diquarks are in the (at least first) excited state. In that case diquark becomes more massive, MD∗ = 2mq − |E2c |, corresponding to the 2sstate. Thus, MD∗ in the QDQ model turns out to be MD∗ = 939.8 MeV and Eq. (8b) now reads as Mζ = 2MD∗ − E1c + mc¯ − E1v . (9) (or δ ) for the given B From here the values of Bov oc v (= 0.8 fm) work out to be 0.27 fm (δc = 0.86). This is indeed much smaller than Boc which in turn implies that the c¯ quark is sitting near the center. This fact is also reflected in the charge distribution of ζ 0 (cf. Fig. 3). Thus for ζ 0 there appears to be a large
Fig. 4. Form factor of ζ 0 (3099) with constituent diquarks in the excited state.
negative charge in the central region and the positive charge remains in the outer periphery. Note that corresponding to the 2s-diquark, the diquark form factor in the first term of (6) should now be read as 2 )(1 − 2Q2 b 2 ) (1 − Q2 boc oc , FD,2 Q2 = 2 )4 (1 + Q2 boc
(10)
which actually corresponds to the diquark charge distribution in the 2s-state, viz., 1 r 2 exp(−r/bo ). 2 − ρD,2 (r) = (11) 32πbo3 bo The form factor of ζ 0 (3099) obtained from (6) is displayed in Fig. 4. For the sake of curiosity, we also demonstrate the charge distribution and the charge form factor of θ + (1540) corresponding to the 2sdiquark in the core in Figs. 5 and 6, respectively. Finally, we wish to append some remarks on the magnetic moments of these pentaquark baryons in this oversimplified QDDQ model. If we assume that the contribution to the magnetic moment of these pen-
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3. Form factor of the tetraquark systems
Fig. 5. Charge distribution of θ + (1540) with constituent diquarks in the excited state.
As concerted efforts are already going on currently at the experimental level to look for possible fourquark bound states, we are tempted to extend the above results to the case of these tetraqurak mesons. We consider these tetraquark mesons as bound states of a di¯ that in fact furnishes quark (D) and an antidiquark (D) a color singlet state, in consonance with our expectation. It is now a matter of straightforward extension of the above results to the case of the diquark (ud) and ¯ constituting the new exotic mesons. antidiquark (u¯ d) In fact the form factor for these exotic mesons, after using the results of KK for the diquark, is given by ¯ 2 D GD E Q =
FD,1 (Q2 ) 1 + 14 Q2 BoDD¯
2
where FD,1 (Q2 ) is given by (2b). Naturally, the cou¯ pling (and hence the Bohr radius BoDD¯ ) for the DD system here is expected to be different from that for the DD system of QDDQ model. The results are presented for the charge distribution in Fig. 7 and the charge form factor in Fig. 8. Fig. 6. Form factor of θ + (1540) with constituent diquarks in the excited state.
4. Multiquark configurations in pentaquark systems taquark baryons comes mainly from the spin-1/2 constituents, then it is the valence quark that will con¯ Thus tribute to it. In θ + it is s¯ whereas in ζ 0 it is c. we expect [17] the magnetic moments to be given as µθ + = +(Mθ /ms¯ ) nm ∼ = 3.12 nm, µζ 0 = −(Mζ /mc¯ ) nm ∼ = −1.94 nm. In the absence of any experimental information on magnetic moments of these baryons it is difficult to draw any definite conclusions regarding the relative magnitudes of the normal and anomalous contributions to the above values of µθ and µζ . It is equally difficult to estimate the role played by the core and the valence in these numerical values. It is, however, abundantly clear that the anomalous part is going to be quite significant for these systems.
With a view to describe various properties of 3quark baryons several models with varying quark configurations (such as sea-valence model, QDQ model, bag or cloudy bag model, etc.) have been used in the past. The problem is, however, more complicated with pentaquark systems. While more experimental results in near future on these systems will act as the guiding principle in this regard, the modest attempt made here in terms of QDDQ model is based on the work of KK [1] and it also embodies the spirit of the recent suggestions of Jaffe and Wilczek [14]. Baryon resonances with strangeness +1 (denoted by Z ∗ ) have also been studied by Jennings and Maltman [18]. They have explored the implications of a number of models compatible with the Z ∗ resonances in the region of θ + (1540) for the as-yet-undetermined quantum numbers of θ + . According to them θ + is the lowest resonance in this region with a width [19] Γ < 6 MeV.
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Fig. 7. Charge distribution of the diquark–antidiquark mesons (a case of tetraquark systems).
boson exchange between the constituent quarks and (ii) when the qq interactions are generated by an effective color-magnetic exchange. No doubt, the predictions are made corresponding to both the “schematic” and the “realistic” situations with regard to the spacedependence of the interactions, but an account of the two-body energies (the ones attributed to the diquark– diquark interactions) is not quite transparent in these calculations. On the other hand, if diquark configurations have to play certain important role (and they have to, as they are also the colored objects) then an account of these two (and perhaps higher)—body energies is necessary. In what follows we shall briefly outline our recent results [20] obtained schematically for these energies in the context of diquark clustering in quark–gluon plasma [21]. While details will appear elsewhere, this is mainly to get a glimpse in to the relative contributions arising from different spin-color diquark states. Evidently, this offers a clue towards a better comprehension of the underlying mechanism in these pentaquark systems. As a matter of fact, starting with the following type of color hyperfine interaction Hamiltonian i · λ j σi · σj , λ Hint = −A i,j
Fig. 8. Form factor of the diquark–antidiquark mesons (a case of tetraquark systems).
While a unified description for both θ + and ζ 0 is suggested in this Letter, a link of QDDQ configuration with the diquark–triquark configuration of θ + proposed by Karliner and Lipkin [10] can easily be established. A straightforward spin-analysis of these pentaquark baryons reveals that the double-diquark system spends 50% of its time in spin-1 configuration. The physical mechanism responsible for this possibility could be that in the vicinity of an antiquark (¯s in θ + and c¯ in ζ 0 ) a spin-0 (ud) diquark gets polarized, mainly because of the increase in the color hyperfine binding energy, presumably as a result of spin-flip. This fact is further corroborated by our recent studies [18] on the diquark clustering in quark– gluon plasma. For the pentaquark systems Jennings and Maltman [18] use the “hyperfine” interactions and account for two cases, namely, (i) when the spin-dependent qq interactions are generated by an effective Goldstone
where A is the strength of interaction and σ ’s and λ’s are the usual Pauli and Gell-Mann matrices, respectively, the interaction energies are computed for all permissible spin-color states of a two-diquark system in the cluster of quarks and in the light of a generalized Pauli Principle. The results are summarized in Table 1 for various spin-color states of two diquarks. Notice that the calculations yield the interaction energy as (14A/3) for the spin-flip case which is larger than the maximum value (4A) obtained for the nonspin-flip case. This is true especially for color states
, |3a ≡ , |15sa ≡ |3s ≡ . However, for other color states, namely,
|6as ≡ , , |15s ≡
, |15as ≡ , |6s ≡
R.S. Kaushal et al. / Physics Letters B 600 (2004) 215–222
Table 1 Two-body (diquark–diquark) interaction energies in multiquark systems in different spin–color states Spin states Color states
|1a
|5
|3s
|1s
|3a
|3a , |15sa , |3s
3A
4A
2A
A
14 A 3
|6as , |6s , |15as
5A 2
2A
13 A 3 20 A 3
37 A 6 34 A 3
3A
|15s
2A 3 − 38 A
4A 3
the situation is found to be different. Thus, the former case of spin-flip seems to favour the physical mechanism mentioned above.
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In the absence of any concrete experimental information available on the properties of θ + and ζ 0 particles studied in this Letter, no meaningful comparison is therefore possible for these predictions. However, the results presented in this communication are at best at the level of speculation, nevertheless they can serve as a useful guide for the future when experimental situation becomes free from ambiguities. A study of the QDDQ model for Ξ −− (1860) particle in terms of its (dsds u) ¯ constituents is in progress.
Acknowledgements 5. Concluding discussion In analogy with the QDQ model for nucleons, we have ventured in this Letter to look into the structural details of the newly discovered exotic pentaquark baryons by way of proposing a QDDQ model for these systems. Extended character of the diquark is appropriately accounted for in predicting the charge distribution and charge form factor for θ + (1540) and ζ 0 (3099) particles. While the results obtained are consistent for the θ + baryon, some abnormal behavior is observed for the ζ 0 baryon in the sense that the heavy valence c¯ in this case is found to remain well within the core and is, therefore, capable to interact only with the excited diquarks in the core. As a matter of fact similar speculations have been made by Jaffe and Wilczek [14]. Within this framework, the magnetic moments for θ + and ζ 0 are predicted to be +3.12 nm and −1.94 nm, respectively. In fact an attempt has been made in the preceding section to find a plausible basis for these double diquark configurations, which in turn couple with the remaining (fifth) antiquark. Based on the situation analogous to other mesons, which are basically the bound states of a quark and an antiquark, we have sought here to expand the ambit to incorporate the possibility of studying the multiquark (four-quark or tetraquark) mesons as bound state of a diquark and an antidiquark. The predictions are made for the charge distribution and the form factor for the ¯ system within this scheme. Of course, bound (ud)(u¯ d) the extended character of the diquark has also been included in these calculations.
Authors wish to thank Professor V.S. Bhasin for several useful discussions. One of us (A.K.S.) gratefully acknowledges the Senior Research Fellowship of CSIR, New Delhi.
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