# Narrow , , and states of in a Quark model with Antisymmetrized Molecular Dynamics

###### Abstract

The exotic baryon is studied with microscopic calculations in a quark model by using a method of antisymmetrized molecular dynamics. We predict that three narrow states, , , and nearly degenerate with the lowest state in the system. We discuss decay widths and estimate them to be for the , and MeV for the state. In contrast to these narrow states, the states should be much broader. We assign the observed as the .

## I Introduction

The exotic baryon has recently been reported by several experimental groups[1, 2, 3, 4, 5, 6, 7, 8, 9]. Since the quantum numbers determined from its decay modes indicate that the minimal quark content is , these induced experimental and theoretical studies of multiquark hadrons. However it should be kept in mind that the has not been well established yet because of the low statistics and experimental reports[10, 11, 12] for no evidence of the .

The prediction of a state of by a chiral soliton model [13] motivated the experiments of the first observation of [1]. Their prediction of even parity is unnatural in the naive quark model, because the lowest state is expected to be spatially symmetric and have odd parity due to the odd intrinsic parity of the anti-quark. Theoretical studies were done to describe by many groups[14, 15, 16, 17, 18, 19, 20, 21], some of which predicted the opposite parity, [18, 19, 20]. The problem of spin and parity of is not only open but also essential to understand the dynamics of pentaquark systems. To solve this problem, it is crucial to calculate five-quark system relying on less a priori assumptions such as the existence of quark clusters or the spin parity.

In this paper we would like to clarify the mechanism of the existence of the pentaquark baryon and predict possible narrow states. We try to extract a simple picture for the pentaquark baryon with its energy, width, spin, parity and also its shape from explicit 5-body calculation. In order to achieve this goal, we study the pentaquark with a flux-tube model[24, 25] based on strong coupling QCD, by using a method of antisymmetrized molecular dynamics (AMD)[22, 23]. In the flux-tube model, the interaction energy of quarks and anti-quarks is given by the energy of the string-like color-electric flux, which is proportional to the minimal length of the flux-tube connecting quarks and anti-quarks at long distances supplemented by perturbative one-gluon-exchange (OGE) interaction at short distances. For the system the flux-tube configuration has an exotic topology, Fig.1(c), in addition to an ordinary meson-baryon topology, Fig. 1(d), and the transition between different topologies takes place only in higher order of the strong coupling expansion. Therefore, it seems quite natural that the flux-tube model accommodates the pentaquark baryon. In 1991, Carlson and Pandharipande studied exotic hadrons in the flux-tube model[26]. They calculated for only a few states with very limited quantum numbers and concluded that pentaquark baryons are absent. We apply the AMD method to the flux-tube model. The AMD is a variational method to solve a finite many-fermion system. This method is powerful for the study of nuclear structure. One of the advantages of this method is that the spatial and spin degrees of freedom for all particles are independently treated. This method can successfully describe various types of structure such as shell-model-like structure and clustering (correlated nucleons) in nuclear physics. In the application of this method to a quark model, we take the dominant terms of OGE potential and string potential due to the gluon flux tube. Different flux-tube configurations are assumed to be decoupled. Since we are interested in the narrow states, we only adopt the confined configuration given by Fig.1(c). We calculate all the possible spin parity states of system, and predict low-lying states. By analysing the wave function, we discuss the properties of and estimate the decay widths of these states with a method of reduced width amplitudes.

## Ii formulation

In the present calculation, the quarks are treated as non-relativistic spin- Fermions. We use a Hamiltonian as follows,

(1) |

where is the kinetic energy of the quarks, represents the short-range OGE interaction between the quarks and is the energy of the flux tubes. For simplicity, we take into account the mass difference between the quarks and the quark, only in the mass term of but not in the kinetic energy term. Then, is represented as follows;

(2) |

where is the total number of quarks and is the mass of -th quark, which is for a or quark and for a quark. denotes the kinetic energy of the center-of-mass motion.

represents the short-range OGE interaction between quarks and consists of the Coulomb and the color-magnetic terms,

(3) |

Here, is the quark-gluon coupling constant, and is defined by , where is the generator of color , for quarks and for anti-quarks. The usual function in the spin-spin interaction is replaced by a finite-range Gaussian,

In the flux-tube quark model [24], the confining string potential is written as , where is the string tension, is the minimum length of the flux tubes, and is the zero-point string energy. depends on the topology of the flux tubes and is necessary to fit the , and potential obtained from lattice QCD or phenomenology. In the present calculation, we adjust the to fit the absolute masses for each of three-quark and pentaquark.

For the meson and 3-baryon systems, the flux-tube configurations are the linear line and the -type configuration with three bonds and one junction as shown in Fig.2(a) and (b), respectively. The string potential given by the -type flux tube in a -baryon system is supported by Lattice QCD [27]. For the pentaquark system, the different types of flux-tube configurations appear as shown in Fig. 1.(e),(f), and (d), which correspond to the states, , , and , respectively. ( is defined by color anti-triplet of .) The flux-tube configuration (e) or (f) have seven bonds and three junctions, while the configuration (d) has four bonds and one junction. In principle, besides these color configurations ( and ), other color configurations are possible in totally color-singlet systems by incorporating a color-symmetric pair as in Refs.[16, 21]. However, since such a string from the is energetically excited and is unfavored in the strong coupling limit of gauge theories as shown in Ref.[28]. Therefore, we consider only color- flux tubes as the elementary tubes. In fact, the string tension for the color-6 string in the strong coupling limit is times larger than that for the color- string from the expectation value of the Casimir operator. The string potentials given by the tube lengths of the configuration Fig.1(c) is supported by Lattice QCD calculations [29].

In the present calculation of the energy, we neglect the transition among , and because they have different flux-tube configurations. It is reasonable in the first order approximation, as mentioned before. In each tube configuration, the minimum length is given by a sum of the lengths() of bonds ( is the number of the bonds. See Fig.2). Here we define to be the length of the path between -th (anti)quark and -th (anti)quark along the flux tubes. For example, in case of the state shown in Fig.2(c), the path lengths are given by the bond lengths as , , , etc. Then we can rewrite in the expectation values of the string potential with respect to a meson system(), a three-quark system(), and the pentaquark states , , , as follows:

(4) | |||||

(5) | |||||

(6) | |||||

(7) |

In the practical calculation, we approximate the minimum length of the flux tubes by a linear combination of two-body distances between the -th (anti)quark the -th (anti)quark as,

(8) | |||||

(9) | |||||

(10) | |||||

(11) |

It is clear that the above equations are obtained by approximating the path length with the distance as for all and pairs. In the meson system, it is clear that Eq.8 gives the exact value. The approximation, Eq.9, for -baryons is used in Ref.[24] and has been proved to be a good approximation. We note that the confinement is reasonably realized by the approximation in Eq.10 for as follows. The flux-tube configuration (e)(or (f)) consists of seven bonds and three junctions. In the limit that the length() of any -th bond becomes much larger than other bonds, the string potential approximated by Eq.10 behaves as a linear potential . It means that all the quarks and anti-quarks are bound by the linear potential with the tension . In that sense, the approximation in Eq.10 for the connected flux-tube configurations is regarded as a natural extension of the approximation(Eq.9) for -baryons. It is convenient to introduce an operator . One can easily prove that the above approximations, 8,9,10,11, are equivalent to within each of the flux-tube configurations because the proper factors arise from depending on the color configurations of the corresponding (or ) pairs.

In order to see the accuracy of the approximations Eqs.9 and 10, we calculate the ratio of the approximated length to the exact in a simple quark distribution with Gaussian form which imitates the model wave function of the present calculation. Figure 3 shows the ratio in a system and a system. The quark positions are randomly chosen in Gaussian deviates with the probability , and values for 1000 samplings are plotted. We use the same size parameter as that of the single-particle Gaussian wave function in the present model explained later. Comparing Figs.3(a) with 3(b), we found that the ratio for the system is about 10% smaller than that for the system. Since the zero-point energy in the string potential is adjusted in each of the and the , this underestimation should relate only to the relative energy of the string potential in each system, and may give a minor effect on the level structure of the pentaquark.

We solve the eigenstates of the Hamiltonian with a variational method in the AMD model space[22, 23]. We take a base AMD wave function in a quark model as follows.

(12) | |||

(13) |

where is the parity projection operator, is the anti-symmetrization operator, and the spatial part of the -th single-particle wave function given by a Gaussian whose center is located at in the phase space. is the spin-isospin-color function. For example, in case of the proton, is given as . Here, is the intrinsic-spin function and expresses the color function. Thus, the wave function of the quark system is described by the complex variational parameters, . By using the frictional cooling method [22] the energy variation is performed with respect to .

For the pentaquark system (,

(14) |

where and correspond to the configurations and in Fig.1, respectively. Since we are interested in the confined states, we do not use the meson-baryon states, . This assumption of decoupling of the reducible and irreducible configurations of the flux tubes can be regarded as a kind of bound-state approximation. The decoupling of the different flux-tube configurations can be characterized by the suppression factor from the transition of the gluon field in the non-diagonal matrix elements . In a simple flux-tube model, is roughly estimated by the area swept by the tubes when moving from one configuration into the other configuration as . We make an estimation of the expectation value of by assuming a simple quark distribution with Gaussian form which imitates the model wave function in the same way as the evaluation of the . The suppression factor among the configurations , , and is estimated to be within the present model space. Therefore, we consider that the present assumption of the complete decoupling in the energy variation is acceptable in first order calculations.

The coefficients for the spin function are determined by diagonalization of Hamiltonian and norm matrices. After the energy variation with respect to the and , the intrinsic-spin and parity eigen wave function for the lowest state is obtained for each . In the AMD wave function, the spatial wave function is given by multi-center Gaussians. When the Gaussian centers are located in some groups, the wave function describe the multi-center cluster structure and is equivalent to the Brink model wave function(a cluster model often used in nuclear structure study) [30, 31]. On the other hand, because of the antisymmetrization, it can also represent shell-model wave functions when all the Gaussian centers are located near the center of the system[30, 31]. In nuclear structure study, it has been already proved that the AMD is one of powerful tools due to the flexibility of the wave function[23]. In general, the relative motions in the AMD are given by such Gaussian forms as where is a Jacobi coordinate and is given by a linear combination of the Gaussian centers . Here we explain the details of the relative motion in a simple case of a two-body cluster structure in a system. If the Gaussian centers are located in two groups as and , and if each group does not contain identical particles, the wave function expresses the two-body cluster state, where each cluster is the harmonic oscillator -orbital state, , with zero orbital-angular momentum. The inter-cluster motion is given as , where , and is the relative coordinate between the clusters. In the partial wave expansion of the inter-cluster motion ,

(15) |

where is the modified spherical Bessel function, it is found that the wave function contains higher orbital-angular momentum components in general. However, in case of , the wave function is dominated by the lowest component since the components rapidly decrease with the increase of . As a result, the even-parity and odd-parity states are almost the eigen states, while the odd-parity and even-parity states are nearly the eigen states. (The contains an odd intrinsic parity of the in addition to the parity of the spatial part.) Therefore, we do not perform the explicit -projection in present calculation for simplicity. We have actually checked that the obtained wave functions are almost the -eigen( or ) states and higher components are minor in most of the and states.

In the present wave function we do not explicitly perform the isospin projection, however, the wave functions obtained by energy variation are found to be approximately isospin-eigen states in most of the low-lying states of the and due to the color-spin symmetry.

In the numerical calculation, the linear and Coulomb potentials are approximated by seven-range Gaussians. We use the following parameters,

(16) |

Here, the quark-gluon coupling constant is chosen so as to fit the and mass difference. The string tension is adopted to adjust the excitation energy of . The size parameter is chosen to be fm.

## Iii Results

In table.1, we display the calculated energy of states with (), (), (). The zero-point energy of the string potential is chosen to be MeV to fit the masses of systems, , and . The calculated masses for with and correspond to the experimental data of and . The contributions of the kinetic and each potential terms are consistent with the results of the Ref.[26]. We checked that the obtained states are almost eigen states of the angular momentum and the projection gives only minor effects on the energy.

Kinetic() | 1.74 | 1.87 | 1.66 | 1.93 | 2.09 |
---|---|---|---|---|---|

String() | 0.02 | 0.27 | 0.07 | 0.03 | 0.25 |

Coulomb | 0.65 | 0.52 | 0.62 | 0.65 | 0.53 |

Color mag. | 0.17 | 0.09 | 0.14 | 0.16 | 0.14 |

0.94 | 1.52 | 1.24 | 1.14 | 1.67 | |

exp. (MeV) |

Now, we apply the AMD method to the system. For each spin parity, we calculate energies of the and states and adopt the lower one. In table.2, the calculated results are shown. We adjust the zero-point energy of the string potential as MeV to fit the absolute mass of the recently observed . This for pentaquark system is chosen independently of for -baryon. If is assumed as Ref.[26], the calculated mass of the pentaquark is around 2.2 GeV, which is consistent with the result of Ref.[26].

The most striking point in the results is that the and states nearly degenerate with the states. The correspond to (), and the is (). The lowest () state appears just below the and the second () state is at the same energy as the (,) states. However these states, as we discuss later, are expected to be much broader than other states. The and exactly degenerate in the present Hamiltonian which does not contain the spin-orbit force. Other spin-parity states are much higher than these low-lying states.

Kinetic() | 3.23 | 3.22 | 3.36 | 3.19 | 3.19 | 3.36 | 3.33 |
---|---|---|---|---|---|---|---|

String() | 0.66 | 0.55 | 0.64 | 0.64 | 0.56 | 0.54 | |

Coulomb | 1.05 | 1.04 | 0.99 | 1.03 | 1.03 | 0.99 | 0.98 |

Color mag. | 0.01 | 0.25 | 0.04 | 0.19 | 0.06 | 0.17 | |

Color mag. | 0.01 | 0.00 | 0.02 | 0.06 | 0.02 | 0.04 | |

1.50 | 1.53 | 1.56 | 1.56 | 1.71 | 1.75 | 1.98 |

## Iv discussion

In this section, we analyze the structure of the obtained low-lying states of the system, and discuss the level structure and the width for decays.

### iv.1 Structure of low-lying states

We analyze the spin structure of these states, and found that the states consist of two spin-zero -pairs, while the contains of a spin-zero -pair and a spin-one -pair. Here we call the color anti-triplet pair with the same spatial single-particle wave functions as a -pair and note a spin -pair as . Since the -pair has the isospin and the -pair has the isospin because of the color asymmetry, the state is isovector while the lowest even-parity states are isoscalar. The state corresponds to the (1530) in the flavor -plet predicted by Diakonov et al.[13]. It is surprising that the odd-parity state, has the isospin , which means that this state is a member of the flavor -plet and belong to a new family of baryon. We denote the states by , and the state by . The mass difference is about 30 MeV. In the energy region compatible to the and states, there appear two states. The lowest one is the state with and pairs, while the higher one is the with and pairs. The former is the isospin symmetric state and is dominated by component. The latter is isovector and is regarded as the spin -partner of the state. The state is the lowest in the system. We, however, consider this state not to be the observed because its width should be broad as discussed later.

Although it is naively expected that unnatural spin parity states are much higher than the natural spin-parity state, the present results show the abnormal level structure of the system, where the high spin state and the unnatural parity states nearly degenerate just above the state. By analysing the details of these states, the abnormal level structure can be easily understood with a simple picture as follows. As shown in table.2, the states have larger kinetic and string energies than the and states, while the former states gain the color-magnetic interaction. It indicates that the degeneracy of parity-odd states and parity-even states is realized by the balance of the loss of the kinetic and string energies and the gain of the color-magnetic interaction. In the and the states, the competition of the energy loss and gain can be understood by Pauli principle from the point of view of the -pair structure as follows. As already mentioned by Jaffe and Wilczek[14], the relative motion between two -pairs must have the odd parity () because the is forbidden between the two identical -pairs due to the color antisymmetry. In the state and the second state, one of -pairs is broken to be a -pair and the is allowed because two diquarks are not identical. The is energetically favored in the kinetic and string terms, and the energy gain cancels the color-magnetic energy loss of a -pair. Also in the lowest state, the competition of energy loss and gain is similar as each contribution of the kinetic, string and potential energies in the lowest state is almost the same as those in the and the second (table 2). It means that the gain of the kinetic energy of the state compete with the color-magnetic energy loss in the lowest as well as the and the second .

We should stress that the existence of two spin-zero -diquarks in the states predicted by Jaffe and Wilczek[14] is actually confirmed in the present calculations without a priori assumptions for the spin and spatial configurations. In fact, the component with two spin-zero -pairs is 97% in the present state. In Fig.4, we show the quark and anti-quark density distribution in the states and display the centers of Gaussians for the single-particle wave functions. In the intrinsic wave function, Gaussian centers for two -pairs are located far from each other with the distance about 0.6 fm. It indicates the spatially developed diquark-cluster structure, which means the spatial and spin correlations in each -pair. It is found that the center of the stays at the same point of that of one , as and where are the Gaussian centers in Eq.12 and fm. As a result, we found the spatial development of - clustering and a parity-asymmetric shape in the intrinsic state before parity projection(Fig.4). As explained in II, the wave function is equivalent to the - cluster wave function in Brink model[30] with relative motion. After the parity projection, the is exchanged between two diquarks. In contrast to the spatially developed cluster structure in the even-parity state, the odd-parity states are almost the spatially symmetric states with spherical shapes.

As mentioned before, the degeneracy of the even-parity states and the odd-parity states originates in the balance of the excitation energy and the energy gain of the color-magnetic interaction. Here we consider the excitation energy as the total energy lose in the kinetic, string and Coulomb terms. It is important that GeV in the pentaquark is much smaller than GeV in the nucleon system. The reason for the relatively small in the pentaquark can be easily understood by the - cluster structure. In the two-body cluster state with the relative motion, the is roughly estimated by the reduced mass of two clusters, as is given as ( and are the masses of the clusters). In the nucleon, is obtained from the - cluster structure in the state, while for the pentaquark system is found in the - clustering. The reduced mass in the pentaquark is times larger than that in the nucleon system, therefore, should be smaller in the pentaquark than in the nucleon by the factor . This factor is consistent with the present values.

We give a comment on the -splitting between and . In the present calculation, where the spin-obit force is omitted, the and states exactly degenerate. Even if we introduce the spin-orbit force into the Hamiltonian, the -splitting should not be large in this diquark structure because the effect of the spin-orbit force from the spin-zero diquarks is very weak as discussed in Ref.[33].

We remark that the - cluster structure in the present result is different from the diquark-triquark structure proposed by Karliner and Lipkin [16] because the -triquark in Ref.[16] is the with the color-symmetric spin-one -diquark. In the -triquark, the quark should be tightly bound in the triquark due to the strong color-magnetic interaction between and . On the other hand, in the present -cluster, the feels no strong color-magnetic interaction and is bound more weakly than in the -triquark. The color-6 flux tubes are not taken into account in the present framework since they are excited. However, the -triquark might be possible if the short-range correlation in the triquark make the flux-tube short enough to be excited into the color-6 flux-tube.

### iv.2 Width for decays

In the decays, it is important that the allowed decay mode in the () is wave, which should make the state narrower than the () because of higher centrifugal barrier. We estimate the -decay widths of these states by using a method of reduced width amplitudes[31, 32]. This method has been applied for the study of -decay width in the nuclear physics within bound state approximations. In this method, the decay width is estimated by the penetrability of the barrier and the reduced width as a function of the threshold energy and the channel radius ,

(17) |

where is the reduced mass, is the wave number , and () is the regular(irregular) spherical Bessel function. is the probability of decaying particle at the channel radius . We define . In the following discussion, we choose the channel radius fm and MeV. Since the transitions between the different flux-tube configurations, a confined state and a decaying state , are of higher order, the should be small in general when the suppression by the flux-tube transition is taken into account. Here, we evaluate the maximum values of the widths for the states with the method of the reduced width amplitudes, by using meson-baryon probability considering only the simple overlap for the quark wave functions. , then, the decay width can be rewritten in a simple form as

In case of even-parity states, the decay modes are the -wave, which gives MeV fm. By assuming and harmonic-oscillator wave functions for and , we calculate the overlap between the obtained pentaquark wave function and the state. As explained in the previous subsection, the states have the - cluster structure where five Gaussian centers are written as and . We assume a simple wave function as follows,

(18) | |||

(19) |

where the a are chosen as , , and the spin-isospin-color wave function is taken to be

(20) |

The same size parameter as that of the pentaquark is used. The coefficients for the spin function are taken to express the proton and the pseudoscalar meson. The probability is evaluated by the overlap with the obtained wave function,