Understanding penta quark with various quark models

The pentaquark state recently discovered has been studied with three quark models which either fit the nucleon spin structure or the $NN$ scattering. A minimum $\Theta^+$ mass of 1620 MeV is obtained both for the ${1/2}^\pm$ state. The mixing of various color structure configurations, which would reduce the mass of the penta-quark state, should be taking into account in the calculation of penta-quark mass.

with mass∼1540 MeV, and width Γ < 25 MeV. This state is either identified from the decay product nK + or pK s , but up to now no experiment has identified both. In addition, the NA49 collaboration claimed that they found the anti-decuplet partner Ξ −− of Θ + [2]. The HERA-H1 collaboration claimed that they found the charm penta-quark Θ c [3].
These measurements might be contaminated by the normal meson production due to the kinematical reflection [4,5]. The NA49 claim has been challenged by another CERN group based on Ξ spectroscopy data with higher statistics [6]. HERA-B p-nucleus reaction data has not found the Θ + [7]. BES J/Ψ decay data analysis has not found the Θ + either [8].
There are other groups that have not found the Θ + [9].
The results of the reanalysis of the K + d and K 0 L p scattering data are diverse [10,11]. W.R. Gibbs reanalyzed the K + d scattering data, taking into account the double scattering, and found a structure corresponding to a resonance with width of 0.9±0.2 MeV, which is either a 1.547±0.002 GeV 1 2 − or a 1.559±0.003 GeV 1 2 + state and 1 2 − is favored [12]. In addition, a very tiny bump had appeared in 1973 CERN K + p → pK 0 s π + inelastic scattering data [13].
Theoretical studies based on the chiral soliton quark model played an important role in triggering the Θ + searches [14]. In the chiral soliton quark model, the Θ + is a member of the anti-decuplet rotational excitation following the well established octet and decuplet baryons [15,16]. The QCD background of this model has been debated between different groups [17,18,19,20,21,22] Various quark models have been proposed to understand the Θ + , mainly aimed at understanding the parity, low mass and narrow width. In the naive quark model [23] the ground state should be an S-wave one and this means Θ + should be a negative parity state. The S-wave uudds configuration will have S-wave KN components. However both the I = 0, 1 KN S-wave phase shifts are negative in the Θ + energy region [24] and this means that Θ + can not be an S-wave KN resonance, except the new analysis of Gibbs [12]. On the other hand, since the I = 0 KN P -wave P 01 phase shifts are positive, there might be resonance in this channel and this is consistent with the J p = 1 2 + predicted by the chiral soliton quark model. Bag model gave similar result [25,26].
Various quark correlations have been proposed such as color Cooper pair and others [17,27,28] aimed to reconcile the difference between the quark model predictions and the properties of Θ + . It seems to be able to get an even parity penta-quark ground state with small width but hard to get as low as 1540 MeV mass [29]. For example, Yu. A. Simonov did a quantitative calculation based on the Jaffe-Wilczek configuration [17] by means of the effective Hamiltonian approach. The calculated Θ + mass is about 400 MeV higher than the observed 1540 MeV [30]. Different quark interaction mechanisms such as quark-gluon, Goldstone boson exchange and instanton interaction have all been tried to understand the Θ + [17,31,32]. It is fair to say up to now there is no ab initio dynamical calculation to obtain a Θ + mass as low as 1540 MeV with the constraint that these model parameters fit the normal hadron spectroscopy [33,34].
QCD sum rule and lattice QCD both have been used to calculate the penta-quark, the results are diverse. One Lattice QCD group reported they have not observed either I = 0 or I = 1, J p = 1 2 ± bound penta-quark state [35]. Two groups observed odd and one observed even parity state [36]. The pitfalls of these lattice QCD calculations have been discussed in [37]. The Θ + is a resonance state and so its mass should be a complex number. It might be Our group has done three quark model calculations. The first one is an application of the Fock space expansion model which we developed to explain the nucleon spin structure [39]. The naive quark model assumes that the baryon has a pure valence q 3 configuration. This is certainly an approximation. One expects there should be higher Fock components, The nucleon spin structure discovered in polarized lepton-nucleon deep inelastic scattering shows that there are intrinsic non-perturbative sea quark components in nucleon and indeed the nucleon spin structure can be explained by a dynamical model of nucleon where the ground state has about 15% q 3 qq component. This means that even the nucleon ground state might be a mixing of tri-and penta-quark components. In the Θ + mass calculation we assume it is a pure uudds five quark state but with channel coupling. Our results are listed below: Here the calculated mass is in units of MeV and K 8 N 8 means the K and N are both in color octets but coupled to an overall color singlet. The S-wave state has strong channel mixing: the amplitudes of KN, K * N, On the other hand the channel mixing is weak for the P -wave state: The am- Lattice QCD and non-perturbative QCD both show that confinement might be due to gluon flux tube (or gluon string) formation in a quark system. The ground state energy of the gluon field in a qq meson and q 3 baryon can be approximated by a potential [40] L i is the distance between the quark i and the Y-shaped gluon junction. r i is the position of quark i. The first term in Eq. (2) is the color Coulomb interaction and the second term is similar to a linear confinement potential.
Most of constituent quark models use a quadratic or linear potential to model the quark confinement, Here λ a i (a = 1 · · · 8) is the color SU(3) group generator. For a single hadron, qq mesons or q 3 baryons, such a modelling can be achieved by adjusting the strength constant a of the confinement potential. The color factor λ i · λ j gives rise to a strength ratio 1/2 for baryon and meson which is almost the ratio for the minimum length of the flux tube to the circumference of the triangle formed by three valence quarks of a baryon.
How to extend the confinement potential to multiquark systems is an open question.
There is a lattice QCD calculation of the penta-quark potential recently [41]. The ground state energy of the gluon field in a penta-quark with color structure qq (3)s(3)qq (3) can be expressed as Here qq ( for these color structures? It is a question needed to be studied further [42]. A penta-quark state should be a mixing of these color structures. Our first model calculation mentioned above shows that channel coupling reduces the calculated ground state penta-quark mass.
It should be true in general that these different color channel mixing will reduce the ground state energy.
To do a model calculation for a multi-quark system with the above multi-body interaction and multi-channel coupling is numerically quite involved. We have developed a model, called the quark delocalization, color screening model (QDCSM): 1. We re-parameterize the confinement potential Eq.(3) to take into account the effect of channels coupling in multiquark systems induced by various color structures, which are not possible for a qq meson and q 3 baryon; 2. To take into account the orbital excitations but keep the numerical calculation simple we still use quark cluster bases but a delocalized quark orbital wave function is used to allow the multi-quark system to choose its own favorable configuration, i.e., to allow the multi-quark system to vary from the asymptotic hadronic cluster state to a genuine multiquark state and all intermediate configurations [43]. This model explains the existing BB interaction data (bound state deuteron and NN, NΛ, NΣ scattering) well with all model parameters fixed by hadron spectroscopy except for only one additional parameter, the color screening constant µ which is determined by the deuteron properties. More important, it is the unique model, so far, which explains a long-standing fact: The nuclear force and the molecular force are similar except for the obvious difference of length and energy scales; the nucleus is approximately an A nucleon system rather than a 3A quark system [44].  The penta-quark mass has been calculated with this model (QDCSM) in a single color singlet KN channel approximation. As explained before, the effect of the coupling of other color structures and orbital excitations is assumed to have been included in the modelling of QDCSM. In the I = 1 S-wave KN channel, a pure repulsive effective interaction is obtained as shown in Fig.1 (corresponding to curve with µ = 1). This is consistent with the KN scattering data and helps to rule out the I = 1 possibility for the Θ + . For comparison the naive quark model result is also shown in Fig.1 (corresponding to curve with µ = 0), which shows a stronger repulsion. In the I = 0 S-wave KN channel, an effective attraction is obtained and shown in Fig.2 (µ = 1 curve). This is inconsistent with the VPI KN scattering phase shifts [24] but might be consistent with Gibbs new results [12]. A Θ + mass of 1706 MeV is obtained from the minimum of Fig.2 µ = 1 curve. Part of the overestimate of the Θ + mass is due to the overestimate of K mass, which is 650 MeV in this approach. This can be eliminated as follows: One first gets an effective interaction potential from Fig.2  for comparison. It is almost a pure repulsive interaction and will not accommodate a Θ + resonance. The K + n effective potential is shown in Fig.3, which is a very weak repulsive interaction resulted from a cancellation of the I = 1, 0 channel ones. This result shows it is hard to get reliable K + n scattering amplitude from K + d scattering data because it is a small component in comparison to the big K + p amplitude. Gibbs analysis shows the additional complications [12]. To get reliable I = 0 scattering phase shifts from K + n is even harder because one has to get two big ones (corresponding to I = 1, 0 separately), which have opposite sign and so cancel each other, from a small one corresponding to (I = 1) + (I = 0).
For the P -wave channels, we only obtain spin averaged effective KN interactions because the spin-orbit coupling has not been included yet. In the I = 0 channel, there is a strong attraction (shown in Fig.4), as wanted in other quark models with correlations. However in our model the P -wave attraction is not strong enough to overcome the kinetic energy increase to reduce it to be a ground state. This is consistent with the lattice and QCD sum rule results [36,38]. In the I = 1 channel, only a very weak attraction is obtained. This rules out the I = 1 Θ + again.   Fig.6). The Θ + mass is similar to our second model one. Fig.7 shows the result obtained with the naive quark model Hamiltonian, the minimum is 1799 MeV corresponding to a vanishing triangle. Summary: Multi-quark states have been studied for about 30 years. The Θ + , if further confirmed, will be the first example. Once the multi-quark "Pandora's box" is opened, the other multi-quark states: tetra-quark, hexa-quark (or dibaryon), etc., can no longer be kept inside. One expects they will be discovered sooner or later and there are claims that some tetra-quark states have been observed [45,46,47,48]. A new landscape of hadron physics will appear and it will not only show new forms of hadronic matter but will also exhibit new features of low energy QCD.
Nonperturbative and lattice QCD have revealed the color flux tube (or string) structure of the qq, q 3 and even q 4q states. The multi-quark system will have more color structures.
How do these color structures interplay within a multi-quark state? Nuclear structure seems to be understood in terms of colorless nucleons within a nucleus. Multi-quark states might be not so. We emphasized that the effect of non-trivial color structures of multi-quark system should be studied. The low mass and narrow width of the Θ + might be related to such new structures instead of to residual interactions.