# Presentation: 811674

## Anisotropic lattice QCD studies of  penta-quarks and tetra-quarks

_**N. Ishii**_              (Univ. of Tokyo)
- in collaboration with
- T. Doi                (Riken BNL)H. Iida               (TITECH)Y. Nemoto        (Nagoya Univ.)M. Oka              (TITECH)F.  Okiharu        (Nihon Univ.)H. Suganuma    (Kyoto Univ.)K. Tsumura        (Kyoto Univ.)

_**Plan of the talk:**_1 Introduction2 General Formalisms3 Numerical Results4 Summary/Discussion(5 Tetra-quarks(4Q))
- See Phys.Rev.D71,034001(2005); D72,074503(2005) for detail.

**START**

## 1.Introduction

- One of the most important issues for Θ+(1540) is to understand
-                **its **_**extremely narrow decay width**_** Γ<1 MeV.**
- Several ideas have been proposed as
- I=2 assignment
- Jaffe-Wilczek’s diquark picture       ⇒ JP=1/2(+) and 3/2(+)
- πKN hepta-quark picture                 ⇒ JP=1/2(+)
- The string picture
- JP=3/2(-) assignment                       ⇒ JP=3/2(-)

- In this talk, we are mainly interested in _**J**__**P**__**=3/2(±) possibilities**_:
- We first present our numerical results on **J****P****=1/2(±) penta-quarks **brieflyemplyoing **a diquak-type interpolating field**using a flavor dependent boundary condition(**HBC**)
- We then present our numerical results on **J****P****=3/2(±) penta-quarks**employing **three Rarita-Schwinger interpolating fields**using **1000 gauge field** configurations for high statistics

## JP=3/2(±) possibilities

**J****P****=3/2(****-****) assignment can solve **_**the puzzle of the narrow decay width.**_(proposed by A.Hosaka et al., PRD71,074021(2005).)_**Advantage:**_(a) It allows the configuration of **(0s)****5**.(b) It only decays into a **d-wave KN state.****Suppressed overlap** between _these two states_          _**The decay width is expected to be significantly narrow**_._**Disadvantage:**_(a) The constatituent quark picture suggests _**such a 5Q state is quite heavy.**_**        **(← due to the color-magnetic interaction)Since it is not apriori clear whether such a conventional framework can be applied to a new exotic 5Q system as Θ+(1540) or not, _**it is desirable to perform a direct lattice QCD calculation. **_
**J****P****=3/2(+) is also interesting, which is suggested by the diquark picture.**

**J****P****=3/2(±) possibilities**

**HOWEVER**

- The total spin(parity) is **1/2(+) **or **3/2(+).****⇒ ****3/2(+) penta-quark **may have** a narrow decay width !**

**p-wave**

## Lattice QCD studies of the penta quarks

_**Lattice QCD studies of the penta quarks**_

**★**** ****There are several lattice QCD calculations of penta-quarks available.**** ****    ****     (published one only)**

**SPIN 1/2**

**SPIN 3/2**

**Most of them are devoted to spin 1/2 states except for the recent two.**

## 2.General Formalism

_**Lattice QCD Setup**_:
- Gauge Config by **standard Wilson gauge**** ****action**:
- Lattice size : **12****3****×96****    **[(2.2fm)3×4.4fm in physical unit]
**β****= 5.75**
- Lattice spacing:                          from Sommer parameter r0.
_**Anisotropic lattice**_**Renormalized anisotropy**: **a****s****/a****t****=4****for **_**accurate measurements of correlators and masses**_
**#(gauge config) = ****504   for J****P****=1/2(±)**                      **= 1000 for J****P****=3/2(±) **
_**O(a) improved**_ Wilson quark **(clover) action**.The quark mass covers the region ms < mq < 2 m s
**Smeared source** to reduce higher spectral contributions

_**2.General Formalism**_

|  | 0.1240 | 0.1230 | 0.1220 | 0.1210 |
| --- | --- | --- | --- | --- |
|   | 656(2) | 784(1) | 893(1) | 1005(1) |
|  | 1011(5) | 1085(4) | 1162(3) | 1240(3) |

## The interpolating fields

_**NK**__*****__**-type**__** **_

_**color-twisted NK**__*****__**-type**__** **_

_**diquark-type **_

_**★**__** **__**Three Rarita-Schwinger interplating fields for **__**J**__**P**__**=3/2(±) states**__**:**_

_**★**__** **__**A diquark-type interplating fields for **__**J**__**P**__**=1/2(±) states**__**:**_

**We consider the following iso-scalar interpolating fields:**

**(scalar)**

**(pseudo scalar)**

**(scalar)**

**(vector)**

## Hybrid Boundary Condition(HBC)

_**Hybrid Boundary Condition(HBC)**_

- We utilize _a flavor dependent spatial BC__** (Hybrid BC (HBC)).**_**               **(We use HBC** **in addition to the _**standard periodic BC(PBC)**_)

|  | quark contents | spatial BC | minimum momentum |  |
| --- | --- | --- | --- | --- |
| N |  | anti-periodic BC |  |  |
| K,K* |  | anti-periodic BC |  |  |
|  |  | periodic BC |  |  |

_**Hybrid Boundary Condition(HBC)**_

| u quark | spatially anti-periodic BC |
| --- | --- |
| d quark | spatially anti-periodic BC |
| s quark | spatially periodic BC |

**Cosequence on hadrons**

**◎** NK and NK* _**threshold energies(s-wave)**_ are **raised** due to              , **◎** _**Θ**__**+**__**,**_if it is a compact resonance, _**will not be affected so much.**_
- HBC can be used to determine whether a state is a compact resonance or not.
**※** In the case of p/d wave, HBC serves as another boundary condition(other than PBC).

**With HBC**

## 3.Numerical Results: JP=1/2(±) states (effective mass plots)

- “**Effective mass**” is defined as
- which can be considered as an **“weighted average” of  masses**at each time-slice t.

_**J**__**P**__**=1/2(-) state:**_A state appears slightly above the NK threshold (mN+mK).
_**J**__**P**__**=1/2(+) state:**_A state appears above the _raised NK threshold_ (due to the finite box).⇒ **rather massive** !

**Excited state contributions are reducing**

**A single state dominate the correlator G(t) in this region.**

## “Effective mass plot”

_**“**__**Effective mass plot”**_

_**The correlator can be expressed as a sum:**_

_**The Effective Mass is defined as:**_

**negligible !**

_**If G(t) is dominated by a single state:**_

_**Then we have,**_

- This can be considered as_** “average” of masses **_at each time-slice t

**(Constant effective mass)**

## Chiral extrapolation (JP=1/2(±))

_**At physical point**_
- (1) JP=1/2(+):   **2.24(11) GeV**(2) JP=1/2(-):   **1.75(3)    GeV**

**NK threshold (p-wave)**

**NK threshold (s-wave)**

- Our data **does not support** a low-lying **J****P****=1/2(+) penta-quark****.**
- For **J****P****=1/2(-) state**, the mass(1.75 GeV) is OK !Still, it is necessary to check whether it is not an NK scattering state but a compact resonance.⇒ **HBC analysis**

## HBC analysis (JP=1/2(-) state)

**PBC**

**HBC**

- NK(s-wave) threshold is raised up by 210 MeV.
- The best fit mass m5Q  is raised up by a similar amount.
- ★ No compact 5Q resonance exists in the region:
- ★ **The state observed in J****P****=1/2(-) is an NK scattering state.**

## Combining the results from all the other hopping parameters.

_**Combining the results from all the other hopping parameters.**_

-  _**data points**_  The best fit value over the   plateau.
-  _**solid lines**_**  **NK(s-wave) threshold

- The states observed in                     are NK scattering states !

## Numerical Results: JP=3/2(-) state (effective mass plot)

_**Numerical Results: **__**J**__**P**__**=3/2(-)**__** state (effective mass plot)**_

**This correlator is too noisy !****Fit is not performed.**

- The **plateaus **appear**above the NK*-threshold ****and**** ****above the raised NK threshold**.

**plateau**

**plateau**

- twisted

**×**

## NK*(JP=3/2(-))

**NK*(s-wave) threshold   is raised up by 179 MeV.**
**NK(d-wave) threshold    is lowered down by 66 MeV.**
**Best-fit(m****5Q****) is raised up by 80 MeV.**Its value is almost consistent with NK*-threshold(s-wave). **This state is an NK* scattering state.**A large number of  config. **Nconf=1000** has played a crucial role.

## color-twisted NK*(JP=3/2(-))

**The situation is similar to the NK*-correlator.**
**NK*(s-wave) threshold   is raised up by 179 MeV.**
**NK(d-wave) threshold    is lowered down by 66 MeV.**
**Best-fit(m****5Q****) is raised up by 90 MeV.**Its value is almost consistent with the NK*-threshold.
-         **This state is also an NK* scattering state.**

- twisted

- twisted

## Chiral extrapolation (JP=3/2(-))

_**○**__**(circle)**_**      **from NK*-type correlator
_**□**__**(box)**_**      **from color-twisted NK*-type correlator

**Physical quark mass region**

**In the physical quark mass region**
**NK*-type:****                       ****m****5Q****= 2.17(4) GeV**
**Color-twisted NK*-type: m****5Q****= 2.11(4) GeV**
**No evidence for a low-lying 5Q state**

- HBC analysis suggeststhese states are** ****NK*(s-wave) scattering states**

- Due to the limited time, we cannot show HBC analysis.

## JP=3/2(+) state (effective mass plot)

_**J**__**P**__**=3/2(+)**__** state (effective mass plot)**_

- The **plateaus **appear**above the raised NK*-threshold ****and**** ****above the raised NK threshold**.

**plateau**

- twisted

**plateau**

**plateau**

## NK*(JP=3/2(+))

_**NK*(J**__**P**__**=3/2(+))**_

- Changes in the two-particle spectrum are too small in JP=3/2(+) channel.
**N*K*(s-wave) threshold is raised up by 170 MeV.**
**NK*(p-wave) threshold   is lowered down by 57 MeV.**
**NK(p-wave) threshold    is lowered down by 66 MeV.**
**Best-fit(m****5Q****) is raised up by 60 MeV.**Its value coincides with N*K*(s-wave) threshold.
**This state is a N*K*(s-wave) threshold.**

## Color-twisted NK*(JP=3/2(+))

_**Color-twisted NK*(J**__**P**__**=3/2(+))**_

- Changes in two-particle spectrum are too small in JP=3/2(+) channel.
**NK*(p-wave) threshold   is lowered down by 57 MeV.**
**NK(p-wave) threshold    is lowered down by 66 MeV.**
**Best-fit(m****5Q****) is raised up by 90 MeV.**
**This state is an NK* scattering state.**

- twisted

- twisted

## Diquark-type(JP=3/2(+))

- Changes in the two-particle spectrum are too small in JP=3/2(+) channel.
**NK*(p-wave) threshold   is lowered down by 57 MeV.**
**NK(p-wave) threshold    is lowered down by 66 MeV.**
**Best-fit(m****5Q****) is raised up by 80 MeV.**
**This state is an NK*-scattering state.**

## Chiral extrapolation (JP=3/2(+))

_**Chiral extrapolation (J**__**P**__**=3/2(+))**_

_**○**__**(circle)**_**      **from NK*-type correlator
_**□**__**(box)**_**      **from color-twisted NK*-type correlator
_**△**__**(triangle)**_      from diquark-type correlator

**In the physical quark mass region,**
**NK*-type:****                         ****m****5Q****= 2.64(7) GeV**
**Color-twisted NK*-type:   m****5Q****= 2.48(10) GeV**
**Diquark-type:****                   ****m****5Q****=2.42(6) GeV**
**No evidence for a low-lying 5Q states.**

**Physical quark mass region**

**NK*(p-wave) scattering states**

**N*K*(s-wave) scattering state**

**HBC analysis suggests:**

- Due to the limited time, we cannot show HBC analysis.

## 4. Summary/discussion

**We have studied **_**spin=1/2 and 3/2 penta-quarks**_** by using **_**the anisotropic lattice QCD**_. For acuracy,(a) renormalized anisotropy as /at = 4(b) O(a) improved Wilson (clover) action for quarks(c) smeared source(d) **large number of gauge configurations**: **Ncf=1000 for J****P****=3/2(±)**
**J****P****=1/2(±) **[with a **diquark-type interpolating field**]
**J****P****=1/2(-) state:                                  J****P****=1/2(+) state:**
**HBC analysis **shows that** **_**this state**_** is an NK scattering state.**
**J****P****=3/2(****±****) **[_**A large statistics**_ as Ncf=1000 has played an important role.]
**Three interpolating fields ****(NK*-**type, **color-twisted NK*-**type, **diquark**-type)
_**Only massive states**_ after the chiral extrapolation:**J****P****=3/2(-) state:                                   J****P****=3/2(+) state:**
- HBC analysis suggests that these 5Q states are NK* and N*K* scattering states.
**Following possibilies would be interesting for Θ****+****(1540):**
- Small quark mass effect(and/or elaborate chiral extrapolation)
- Large spatial volume
- Dynamical quarks
- Elaborate interpolating fields to fit the diquark picture
_**πKN hepta-quark picture**_

_**4. Summary/discussion**_

**Too heavy to be identified as Θ****+****(1540****)**

**See for detail:****Phys. Rev. D71,034001 (2005)****Phys. Rev. D72,074503 (2005)**

## 5.Tetra-quarks(4Q): (work in progress)

_**5.Tetra-quarks(4Q): **__(_work in progress)

- A lattice QCD calculation using the **πKN interpolating field(7-body op.)** is difficult.
**πK subsystem**** is much easier to study.**It is a** 4Q system **with** the quantum number of  ****κ****.**
**κ** together with **f****0****(600), f****0****(980), a****0****(980)** forms **the scalar nonet****, **which are candidates of **tetra(4Q) quark** states.

**4Q with “HBC”**

**4Q with PBC**

**0****+**** meson             (chiral extrapolation)**

- Our preliminary result suggests an existence of a _**4Q resonance**__._
**             ****work in progress.**

**ππ threshold(“HBC”)**

**ππ threshold(PBC)**

**PRELIMINARY**

**ππ scattering state**

- Since these tetra-quarks are interesting target in their own right,
- we are currently performing **4Q lattice QCD calculations** from a more general point of view.