# Presentation: 849003

## Multi-Quarks and Two-Baryon Interaction in Lattice QCD 

**T.T.Takahashi** (YITP,Kyoto U.)
_**F. Okiharu**__ (Nihon U.)_,
**T.Doi** (RIKEN-BNL)

**Multi-Quarks and Two-Baryon Interaction ****in Lattice QCD **

## Discoveries of multi-quark candidates

_**Candidates for the pentaqurk (5Q)**_   2002-
** **

- From KEK Web Site

## Scalar meson puzzle

- <Candidates for Tetraquark in Light quark sector  : ***a*****0****(980),  *****f*****0****(980),  *****f*****0****(600) **>
-         Isoscalar Meson below 2 GeV    *** ***_***f***__**0**__**(600),  **__***f***__**0**__**(980)**_**,  *****f*****0****(1370),  **_***f***__**0**__**(1500),  **__***f***__**0**__**(1710)**_
**                                                                                                   ****?            **usual meson  glueball?
**<Particle Data >**
**Suggested**    **quark model-assignment** for some of **the observed light mesons**

**・・・** The classification of the 0++ mesons is **tentative** and the mixing angle uncertain due to large uncertainties in some of the masses. The ***f*****0****(1500)** in the Meson Summary Table is not in this table as it is hard to accommodate in the scalar nonet. **The light scalars *****a*****0****(980), *****f*****0****(980) and *****f*****0****(600) are often considered as **_**meson-meson resonances**_** or **_**four-quark states**_ and are therefore not included in the table.

- Lightest Scalar Meson
- (Quark Model )

-   Ground State of Scalar Meson       ( Quark Model) :: *3**P**0 *

**<Theory>**
- 1977 Jaffe :: Tetraquark model for scalar quark ( *a*0(980)、 *f*0(980)** **)

**σ**

-   **The ****discoveries of these exotic hadrons**** gives motivation of the ****study 4Q and 5Q   **
**   ****systems**** in terms of quark and gluon (****Q****C****D****)**
-   **Lattice QCD simulation** **is ****the first-principle calculation**** and  ****model independent**

## Global strategy 

**Lattice QCD**

- Quark Model
- calculation

-  Hadron mass
-  Hadron matrix element
-  Quark Wave-Function
-  of Hadron

**Hadron Properties**

-  direct measurement

- We are interested in multi-quark properties.
- However, direct measurement of the multi-quark systems is difficult
- to distinguish between resonance and scattering state.
- For further study on the multi-quark structure, quark model study is also indispensable.
- What is the **Hamiltonian of multi-quark system**?
**Interaction between quarks** is one of important quantities
- in order to understand multi-quark system in terms of quarks and gluons.
- ⇒ **It is important for Modeling**
-     Ref) Fl. Stanc and D. O. Riska, Phys. Lett. B575, 242 (2003).
-              Kanada et al., hep-ph/0404144.

## The static potential in multi-quark systems in Lattice QCD

- We review the static potential in        (mesons) and 3Q (baryons) systems first
- Next, we consider the 4Q and 5Q systems as follows;
- 1. **Numerical calculations**
-      Evaluate Inter-quark potential  based on QCD
- 2.  **Determination of the functional form**
-      Compare lattice QCD results with theoretical conjectures
-      e.g. ( **one-gluon-exchange (OGE)****+****linear** ) potential
**The systematic study from the few body system to the many body system leads the **
**deep understanding of quark nuclear physics**

## The static potential in mesons and baryons  

-   For baryons (Recent study)
-    (a) 3Q potential
-     At short distance :: Coulomb term due to the one-gluon-exchange (OGE)
-     At long distance  :: the Y type confinement term based on the flux-tube picture
-    Ref) T.T.Takahashi et al. PRL 86 (2001) 18, PRD65 (2002)114509.

*The static potential in mesons and baryons**  *

- (b)  Abelian Action density
-       Ref) Ichie et. al, Nucl. Phys. A721 (2003) 899.

-  _      __**system (mesons)**_
-   ⇒ Regge trajectory
-   ⇒ heavy quarkonia analysis
-   ⇒ lattice QCD

- Wilson loop

- r

- V(r)

## Theoretical conjecture for tetraquark and pentaquark potentials

-  We consider these types of color-singlet configurations as
-                                                               for 4Q system
-                                                               for 5Q system
-  We consider ( OGE + confinement ) potentials for 4Q and 5Q systems

*Theoretical conjecture **for tetraquark and pentaquark potentials*

-  A :       Coulomb coefficient,
-  sigma: string tension,
-  Lmin:     minimized flux-tube length (we explain in the next slide),
-  C:         constant
-  We compare **the theoretical conjecture** and **the lattice QCD Monte Carlo **
**  ****simulation**

## Confinement term  ~flux-tube picture

-  3Q system

- Lmin :: the total minimal length of the flux-tube
-              linking 3 quarks

-  σ:: The string tension (about 1GeV/fm)

- Λtype flux-tube

-  5Q system

- Multi - Y type flux-tube

- Y type flux-tube

- ・When all angles of the 3Q
-  triangle do not exceed 120°

- ・An angle of the 3Q triangle
-   exceeds 120°

-  In flux-tube picture,
-  minimized flux-tube forms
-  as shown in the left figure

## Formalism for 4Q and 5Q potentials

- *Lattice size 163  X 32 (β=6.0, a~0.1fm)
-    V ~ (1.6fm)3
- *Wilson gauge action  ( quenched )
- *300 gauge configurations
- *smearing method
-  ( in order to enhance the ground state component )
-  We obtain the 4Q and 5Q state with the large ground-state overlap C0
-    ⇒ This ensures accurate measurement of multi-quark potentials

- *The simulation has been done in SU(3)C lattice QCD
- *Tetraquark Wilson loop *W**4Q* and pentaquark Wilson loop *W**5Q*( gauge invariant  )

- Pentaquark Wilson loop *W**5Q*

- Tetraquark Wilson loop *W**4Q*

## Lattice QCD results for 5Q potential

-    anti-quark is located at middle between two junctions
- ・Theoretical multi-Y Ansatz

-  we determine these coefficients from V3Q results
-  A5Q :: coefficient in Coulomb term (=A3Q=0.1366)
- σ5Q  :: string tension            (=σ3Q=0.0460a-2~0.89GeV/fm)
-  C5Q :: constant term                       (=1.57a-1~5/3C3Q)

-  points    :: lattice data
-  curves :: theoretical Ansatz

**Good agreement with multi-Y Ansatz, **
**( OGE + multi-Y linear ) potential**

- There is
- NO adjustable parameter

- V5Q[a-1]

- h[a]

*Lattice QCD results for 5Q potential*

## Lattice QCD results for 4Q potential

-    The 4Q potential does not depend on d,
-         when “h” is small comparison with “d”
-  We observe the level crossing from the 2 meson state
-    to the connected 4Q state
-   Therefore 4Q potential can be described by

*Lattice QCD results for 4Q potential*

- V4Q[a-1]

-  [a]

- V4Q[a-1]

- [a]

- 2 Meson

-  connected 4Q state

**Good agreement with multi-Y Ansatz **
**                                     ****[ (OGE + multi-Y linear)**** ****potential] **

- (1) h>>d (connected 4Q state)

- Q

- Q

## Studies of multi-quark systems in Lattice QCD

- As a next step,
**Nuclear Force** (Interaction between multi-quarks)
- Talk by **T.Doi               **(**Oct.25 Sec.II-3,15:30-**)
-             “Meson-Meson and Meson-Baryon Interactions in Lattice QCD”
**            ****T.T.Takahashi **(**Oct.25 Sec.II-3,15:45-**)
-             “Nuclear Force in Lattice QCD”

-  this talk

## Summary and concluding remarks

**    **(1) The static 4Q and 5Q potentials can be described as
-           **the sum of the Coulomb term and the multi-Y type confinement term**
-           based on the flux tube picture
-      (2) **The universality of the string tension**
-      (3) The Coulomb coefficients
-      (4) We observe **the flip-flop** (recombination of the flux-tube)

*Summary and concluding remarks*

- Ref ) “Study for the Pentaquark Potential in SU(3) Lattice QCD”
-          F.O, H.Suganuma, and T.T.Takahashi, Phys. Rev. Lett. 94 (2005) 192001.
-          “Detailed Analysis of the Tetraquark Potential and Flip-Flop in SU(3) Lattice QCD”
-          F.O, H.Suganuma, and T.T.Takahashi, Phys. Rev. D72 (2005)  014505.