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We study multi-quark systems in lattice QCD. First, we revisit and summarize our accurate mass measurements of low-lying 5Q states with
J = 1/2 and
I = 0 in both positive- and negative-parity channels in anisotropic lattice QCD. The lowest positive-parity 5Q state is found to have a large mass of about 2.24 GeV after the chiral extrapolation. To single out the compact 5Q state from NK scattering states, we use the hybrid boundary condition (HBC), and find no evidence of the compact 5Q state below 1.75 GeV in the negative-parity channel. Second, we study the multi-quark potential in lattice QCD to clarify the inter-quark interaction in multi-quark systems. The 5Q potential
V
_{5Q} for the QQ-
-QQ system is found to be well described by the “OGE Coulomb plus multi-Y Ansatz”: The sum of the one-gluon-exchange (OGE) Coulomb term and the multi-Y-type linear term based on the flux-tube picture. The 4Q potential
V
_{4Q} for the QQ-
system is also described by the OGE Coulomb plus multi-Y Ansatz, when QQ and
are well separated. The 4Q system is described as a “two-meson” state with disconnected flux tubes, when the nearest quark and antiquark pair are spatially close. We observe a lattice-QCD evidence for the “flip-flop”, i.e., the fluxtube recombination between the connected 4Q state and the “two-meson” state. On the confinement mechanism, the lattice QCD results indicate the flux-tube-type linear confinement in multi-quark hadrons. Finally, we propose a proper quark-model Hamiltonian based on the lattice QCD results.

The Multi-quark physics is one of the new interesting fields in the hadron physics. So far, several new particles have been experimentally reported as the candidates of multi-quark hadrons.

At first, the candidates of pentaquark (5Q) baryons were reported: a narrow peak identified as the

and also a charmed pentaquark, the

As the next important stage, the candidates of tetraquark (4Q) mesons were experimentally observed. The X(3872) [

In the theoretical side, the quark model is one of the most popular models to describe hadrons. In the quark model, mesons and baryons are usually described as

(

First, we investigate the mass and the parity of the 5Q system in lattice QCD. As for the parity assignment of the lowest-lying pentaquark, little agreement is achieved even in the theoretical side: the positive-parity assignment is supported by the chiral soliton model [

Second, we study the inter-quark interaction in multi-quark systems in lattice QCD. The inter-quark force is one of the most important elementary quantities in hadron physics. Nevertheless, for instance, no body knows the exact form of the confinement force in the multi-quark systems directly from QCD. In fact, some hypothetical forms of the inter-quark potential have been used in almost all quark model calculations so far. Then, the lattice QCD study of the inter-quark interaction is quite desired for the study of the multi-quark systems. It presents the proper Hamiltonian in multi-quark systems and leads to a guideline to construct the QCD-based quark model. In this paper, to clarify the inter-quark force in the multi-quark system, we study the static multi-quark potential systematically in lattice QCD using the multi-quark Wilson loop. We investigate the three-quark (3Q) potential [

We show in

This paper is organized as follows. In Section 2, we present an accurate mass calculation of low-lying 5Q systems in anisotropic lattice QCD [

There have been many theoretical studies for multi-quark systems in the context of X(3872) and

Also in lattice QCD, there is no consensus on the existence and the parity assignment of the lowest-lying pentaquark system. Two early works supported the negative-parity state for the

In this section, we perform the accurate mass measurement of the 5Q system in anisotropic lattice QCD, and apply hybrid boundary condition [

As a difficulty on the lattice study of multi-quarks, even if a compact multi-quark resonance state exists, there appears a mixture with several multi-hadron scattering states, even at the quenched level. For instance, in the channel of

We use the anisotropic lattice, where the temporal lattice spacing

We use a non-NK-type interpolating field to extract the

would be important and effective. For instance, in Ref. [

We adopt the non-NK-type interpolating field [

for the 5Q state with spin

Equation (1) cannot be decomposed into N and K in the nonrelativistic limit and its coupling to the NK state is rather weak. Hence, the 5Q resonance state

To distinguish compact resonances from scattering states, we have proposed a useful method with the “hybrid boundary condition” (HBC) [

In lattice QCD with the finite spatial volume

scattering state. The HBC imposes the anti-periodic boundary condition for u and d quarks and periodic boundary condition for s quark, while the periodic boundary condition is usually employed for all u, d, s quarks.

In the HBC, the net boundary conditions of both N(uud,udd) and K(

HBC, N and K have minimum momenta

u quark | d quark | s quark | |
---|---|---|---|

HBC | anti-periodic | anti-periodic | periodic |

standard BC | periodic | periodic | periodic |

N (uud or udd) | K ( | ||
---|---|---|---|

HBC | periodic | anti-periodic | anti-periodic |

standard BC | periodic | periodic | periodic |

To generate gluon configurations, we use the standard plaquette action on the anisotropic lattice as [

with

For the quark part, we adopt the

with the quark kernel

where

For the lattice QCD simulation, we use

Now, using anisotropic lattice QCD, we perform the accurate mass measurement of the low-lying 5Q states with

0.1210 | 0.1220 | 0.1230 | 0.1240 | ||
---|---|---|---|---|---|

1.005 (2) | 0.898 (2) | 0.784 (2) | 0.656 (3) | 0.140 | |

1.240 (3) | 1.161 (3) | 1.085 (4) | 1.011 (5) | 0.850 (7) | |

0.845 (2) | 0.785 (2) | 0.723 (2) | 0.656 (3) | 0.530 (4) | |

1.878 (5) | 1.744 (5) | 1.604 (5) | 1.460 (6) | 1.173 (9) |

In

On the other hand, we get a lower mass for the negative-parity 5Q state as

To clarify whether the observed low-lying 5Q state is a compact 5Q resonance

In

As a lattice QCD result, the mass of the 5Q state is largely raised in the HBC in accordance with the NK threshold, which indicates that the lowest 5Q state observed on the lattice is merely an s-wave NK scattering state. In other words, if there exists a compact 5Q resonance

To conclude, our lattice QCD calculation at the quenched level indicates absence of the low-lying compact 5Q resonance

Now, let us consider the physical consequence of the present null result on the low-lying 5Q resonance

First, the present lattice simulation has been done at the quenched level, where dynamical quark effects are suppressed. This quenching effect is not clear and then it may cause the 5Q resonance

Second, we investigated the 5Q state with spin

Third, we have used a localized 5Q interpolating field in this lattice QCD calculation. However, the actual

the

So far, we have performed the direct mass measurement of 5Q states in lattice QCD, where the path integral over arbitrary states is numerically calculated on a supercomputer. In the path-integral formalism, however, it is rather difficult to extract the state information, such as the wave-function of the multi-quark state, and therefore only limited simple information can be obtained in the direct lattice-QCD calculation.

Actually, to distinguish the compact 5Q resonance

Indeed, to get the wave function is very important to clarify the further various properties of the multi-quark state such as the underlying structure and the decay width, which cannot be obtained practically only with the direct lattice-QCD calculation.

Then, apart from the direct lattice-QCD calculation, we have to seek the way to obtain the proper wave function of the multi-quark state. To do so, we need a proper Hamiltonian for the multi-quark system based on QCD. One possible way in this direction is to construct the quark model from QCD, as was mentioned in Section 1. In the next section, we study the inter-quark interaction in multi-quark systems directly from QCD, and aim to construct the QCD-based quark-model Hamiltonian.

In this section, we study the inter-quark interaction in multi-quark systems using lattice QCD [

As for the potential at short distances, the perturbative one-gluon-exchange (OGE) potential would be appropriate, due to the asymptotic nature of QCD. For the long-range part, however, there appears the confinement potential as a typical non-perturbative property of QCD, and its form is highly nontrivial in the multi-quark system.

In fact, to clarify the confinement force in multi-quark systems is one of the essential points for the construction of the QCD-based quark-model Hamiltonian. Then, in this paper, we investigate the multi-quark potential in lattice QCD, with paying attention to the confinement force in multi-quark hadrons.

So far, only for the simplest case of static

with r being the inter-quark distance.

To begin with, we study three-quark (3Q) systems in lattice QCD to understand the structure of baryons at the quark-gluon level. Similar to the derivation of the

within 1%-level deviation [

To demonstrate the validity of the Y-Ansatz, we show in

and

Here, we consider the physical meaning of the Y-Ansatz. Apart from an irrelevant constant, the Y-Ansatz, Equation (6), consists of the Coulomb term and the Y-type linear potential, which play the dominant role at short and long distances, respectively. The Coulomb term would originate from the one-gluon-exchange (OGE) process. In fact, at short distances, perturbative QCD is applicable, and therefore the inter-quark potential is expressed as the sum of the two-body one-gluon-exchange (OGE) Coulomb potential.

The appearance of the Y-type linear potential supports the flux-tube picture [

In usual many-body systems, the main interaction is described by a two-body interaction and the three-body interaction is a higher-order contribution. In contrast, as is clarified by our lattice-QCD study, the quark confinement force in baryons is a genuinely three-body interaction [

In lattice QCD, a clear Y-type flux-tube formation is actually observed for spatially-fixed 3Q systems [

Now, we proceed to multi-quark systems. We first consider the theoretical form of the multi-quark potential, since we will have to analyze the lattice QCD data by comparing them with some theoretical Ansatz.

By generalizing the lattice QCD result of the Y-Ansatz for the three-quark potential, we propose the one-gluon-exchange (OGE) Coulomb plus multi-Y Ansatz [

for the potential form of the multi-quark system. Here, the confinement potential is proportional to the minimal total length

In the following, we study the inter-quark interaction in multi-quark systems in lattice QCD, and compare the lattice QCD data with the theoretical form in Equation (7). Note here that the lattice QCD data are meaningful as primary data on the multi-quark system directly based on QCD, and do not depend on any theoretical Ansatz.

Next, we formulate the multi-quark Wilson loop to obtain the multi-quark potential in lattice QCD [

Similar to the derivation of the

The tetraquark Wilson loop

where

Here,

The multi-quark Wilson loop physically means that a gauge-invariant multi-quark state is generated at

The multi-quark potential is obtained from the vacuum expectation value of the multi-quark Wilson loop:

Here, we briefly summarize the lattice QCD setup in this calculation. For the study of the multi-quark potential, the SU(3) lattice QCD simulation is done with the standard plaquette action at

In this calculation, the lattice spacing a is estimated as

simulation and 300 gauge configurations for the 4Q potential simulation. The smearing method is used for the enhancement of the ground-state component. We here adopt

We study the pentaquark potential

described by the OGE Coulomb plus multi-Y Ansatz, i.e., the sum of the OGE Coulomb term and the multi- Y-type linear term based on the flux-tube picture [

We show in

In

parameter in the theoretical Ansatz apart from an irrelevant constant.) In

In this way, the pentaquark potential

where

We study the tetraquark potential

1. When QQ and

2. When the nearest quark and antiquark pair is spatially close, the 4Q potential

We show in

For large value of h compared with d, the lattice data seem to coincide with the solid curve of the OGE Coulomb plus multi-Y Ansatz,

where

For small h, the lattice data tend to agree with the dotted-dashed curve of the “two-meson” Ansatz, where the 4Q potential is described by the sum of two

Thus, the tetraquark potential

two-meson state. In other words, we observe a clear lattice QCD evidence of the “flip-flop”, i.e., the flux-tube recombination between the connected 4Q state and the two-meson state. This lattice result also supports the flux-tube picture for the 4Q system.

From a series of our lattice QCD studies [

Furthermore, from the comparison among the

and the OGE result of the Coulomb coefficient A as

in Equations (5), (6), (12) and (13).

Here, the OGE Coulomb term is considered to originate from the OGE process, which plays the dominant role at short distances, where perturbative QCD is applicable. The flux-tube-type linear confinement would be physically interpreted by the flux-tube picture, where quarks and antiquarks are linked by the one-dimensional squeezed color-electric flux tube with the string tension

To conclude, the inter-quark interaction would be generally described by the sum of the short-distance two-body OGE part and the long-distance flux-tube-type linear confinement part with the universal string tension

Thus, based on the lattice QCD results, we propose the proper quark-model Hamiltonian

where

It is desired to investigate various properties of multi-quark hadrons with this QCD-based quark model Hamiltonian

We have studied tetraquark and pentaquark systems in lattice QCD Monte Carlo simulations, motivated by the experimental discoveries of multi-quark candidates.

First, we have performed accurate mass calculations of low-lying 5Q states with

Second, we have studied the multi-quark potential in lattice QCD to clarify the inter-quark interaction in multi-quark systems. We have found that the 5Q potential

described by the OGE Coulomb plus multi-Y Ansatz, when QQ and

This paper is based on the unpublished proceeding (hep-ph/0507187, talk by F.O.) at International Workshop on Quark Nuclear Physics, 22-24 Feb 2005. Phoenix Park, Korea. The lattice QCD Monte Carlo calculations were performed on supercomputers at Osaka University and at KEK.

Fumiko Okiharu,Takumi Doi,Hiroko Ichie,Hideaki Iida,Noriyoshi Ishii,Makoto Oka,Hideo Suganuma,Toru T. Takahashi, (2016) Tetraquark and Pentaquark Systems in Lattice QCD. Journal of Modern Physics,07,774-789. doi: 10.4236/jmp.2016.78072