Inter-Quark Potentials in Baryons and Multi-Quark Systems in QCD H. Suganuma, A. Yamamoto, H. Iida, N. Sakumichi (Kyoto Univ.) with T.T.Takahashi (Kyoto.

Inter-Quark Potentials in Baryons and Multi-Quark Systems in QCD

H. Suganuma, A. Yamamoto, H. Iida, N. Sakumichi (Kyoto Univ.)with

T.T.Takahashi (Kyoto Univ.), F. Okiharu (Nihon U.)

Chiral 07, Nov 13-16 2007, RCNP Osaka

Contents1. Three-Quark Potential in SU(3) lattice QCD2. Multi-Quark Potential in SU(3) lattice QCD3. Heavy-heavy-light quark potential and Light-quark effects to the inter-two-quark interaction in baryons (SU(3) lattice QCD and Analytical model calculation)

Inter-quark potential in QCDIn 1979, M.Creutz performed the first application of lattice QCD simulation for the quark-antiquark potential using the Wilson loop.

Since then, the study of the inter-quark force has been one of the central issues in lattice QCD.

Actually, in hadron physics, the inter-quark force can be regarded as an elementary quantity to connect the “quark world” to the “hadron world”, and plays an important role to hadron properties.

In 1999, in addition to the quark-antiquark potential, we performed the first accurate reliable lattice QCD study for the three-quark (3Q) potential, which is responsible to the baryon structure at the quark-gluon level.

Furthermore, in 2005, we performed the first lattice QCD study for the multi-quark potentials, i.e., 4Q and 5Q potentials, which give essential information for the multi-quark hadron physics.

Note also that the study of 3Q and multi-quark potentials is directly related to the quark confinement properties in baryons and multi-quark hadrons.

First, I review the lattice QCD results for static inter-quark potentials.

Quark-antiquark static potential in Lattice QCDM.Creutz (1979,80)

quark anti-quark

Wilson loop

t

T

r

The quark-antiquark potential canbe obtained from the Wilson Loop.

Quark-antiquark static potential in Lattice QCD

r0=0.5fm:unit

quark anti-quark

V(r) = － lim ln<W>T1TT→∞

Wilson loop

t

T

r

The quark-antiquark potential canbe obtained from the Wilson Loop.

M.Creutz (1979,80)

Summarized lattice QCD dataG.S.Bali (2001)

Takahashi, H.S. et al. (2002)JLQCD (2003)


V(r) = － +σrg2

3π1r

r0=0.5fm:unit

quark anti-quarkM.Creutz (1979,80)



The quark-antiquark potential V(r) is well described by Coulomb + Linear Potential. σ≒ 0.89 GeV/fm

At the short distances, the Q-Q potential behaves as the Coulomb-type potential, which is expected from the one-gluon-exchange (OGE) process.


V(r) = － +σrg2

3π1r

r0=0.5fm:unit

g g

quark anti-quark－




At the long distances, the Q-Q potential behaves as a linear arising potential like a “condenser”, which indicates one-dimensional squeezing of the color-electric flux between quark and antiquark.


V(r) = － +σrg2

3π1r

r0=0.5fm:unit

－





V(r) = － +σrg2

3π1r

r0=0.5fm:unit

quark anti-quarkg g

quark anti-quark

One-dimensional squeezing of color flux between q and q－

M.Creutz (1979,80)



quark anti-quark

Wilson loop

t

T

r

Baryonic Three-Quark Potential in Lattice QCD

quark

quark

quark

What Shape of Color Flux?Confining Force?

Before our study, there was almost No lattice QCD study for the Three-Quark Potential.

This is not so trivial especially for quark confining force in baryons at long distance.

PRThis is the cover of

a recent textbook written by Hosaka and Toki.



This is a nice textbook for the introduction toquark-hadron physics



Look! Is this a correct picture for the color flux tube inside baryons ?

But!

Systematical Studies for Three and Multi-Quark Potentials

in Lattice QCD“Detailed Analysis of Tetraquark Potential and Flip Flop in SU(3) Lattice QCD” F. Okiharu, H. Suganuma and T.T. Takahashi

Physical Review D72 (2005) 014505 (17 pages).

“First Study for the Pentaquark Potential in SU(3) Lattice QCD” F. Okiharu, H. Suganuma and T.T. Takahashi

Physical Review Letters 94 (2005) 192001 (4 pages).

“Detailed Analysis of the Gluonic Excitation in the 3Q System in Lattice QCD” T.T. Takahashi and H. Suganuma


“Gluonic Excitation of the Three-Quark System in SU(3) Lattice QCD”T.T. Takahashi and H. Suganuma

Physical Review Letters 90 (2003) 182001 (4 pages).

“Detailed Analysis of the Three Quark Potential in SU(3) Lattice QCD” T.T. Takahashi, H. Suganuma et al.


“Three-Quark Potential in SU(3) Lattice QCD”T.T. Takahashi, H. Suganuma et al.

Physical Review Letters 86 (2001) 18-21.

V3Q(r) = － lim ln<W3Q>T

1TT→∞

t

i j

k

(i, j, k) characterizethe shape of the 3Q triangle.

i j

k


i j

k


i j

k


i j

k


-m l

n

(l, m, n) characterize

the shape of another type of 3Q triangles.

More than 300 different shapes of 3Q triangles are analyzed in total.

Lmin ： total length of string linking three valence quarks

quark

quark

quark

color electric flux


quark

quark

quark


Before our study, there was almost No lattice QCD study for the Three-Quark Potential

Takahashi, H.S. et al. PRL 86 (2001) 18Takahashi, H.S. et al. PRD65 (2002)114509 Takahashi, H.S. PRL 90 (2003) Takahashi, H.S. PRD70 (2004) 074506 Okiharu, H.S. et al. PRD72 (2005) 014505


V3Q(r)conf

quark

quark

quark

color electric flux



V3Q(r)

V3Q(r) = ∑ + σLming2

4πTi

aTja

|ri - rj|i<j

3Lmin ： total length of string linking three valence quarks

One-Gluon-ExchangeCoulomb potential

Linear potential based on string picture

quark

quark

quark

conf

color electric flux


Lattice QCD result for Color Flux-Tube Formation in baryons

H. Ichie et al., Nucl. Phys. A721, 899 (2003)

The status of our studies of 3Q potential

Our studies of the 3Q potential are introduced as “one whole subsection” with citing 4 our papers in 3rd edition of “Lattice Gauge Theories”, which is one of the most popular lattice QCD text books.



?



I have corrected it with the appropriate picture for the color flux tube inside baryons.



I have corrected it with the appropriate picture for the color flux tube inside baryons.

Without a matter of the cover,this is a nice textbook for the introduction to quark-hadron physics.

PRThis is the cover of the proceedings of

Confinement Conference.

Multi-Quark Hadrons and Multi-Quark Multi-Quark Hadrons and Multi-Quark PotentialsPotentials

In these years, there have been reported experimental discoveries of several candidates of multi-quark hadrons such as Θ+(1530), X(3872) and so on.Very recently, the discovery of a “charged charmonium” Z+(4430) (ccud) is reported at KEK-Belle experiment.- -

Tetra-Quark Z(4430) from KEK press Tetra-Quark Z(4430) from KEK press releaserelease

The charged charmonium Z+(4430) is a manifest Tetra-Quark hadron composed by ccud.- -

Multi-Quark Hadrons and Multi-Quark Multi-Quark Hadrons and Multi-Quark PotentialsPotentials

In these years, there have been reported experimentaldiscoveries of several candidates of multi-quark hadrons such as Θ+(1530), X(3872) and so on.Very recently, the discovery of a “charged charmonium” Z+(4430) (ccud) is reported at KEK-Belle experiment.

For the quark-model calculation of the multi-quark system, it is rather important to clarify the multi-quark potential, which gives the quark-model Hamiltonian for multi-quark system.

In fact, the quark model analysis with appropriate multi-quark potential clarifies whether each exotic hadron exists or not, gives the properties of multi-quark hadrons, and predicts new-type exotic hadrons theoretically.

We perform first study of multi-quark potential in lattice QCD.

- -

First Lattice QCD Study for Static Quark Potential in Multi-Quark System

Okiharu, H.S. et al. PRL 94 (2005) 192001 Okiharu, H.S. et al. PRD72 (2005) 014505

quark

quark anti-quark

anti-quark

4 quark system


?


quark

quark

quark

quark

anti-quark

5 quark system


?



quark

quark

quark

quark

quark

quark

anti-quark

anti-quark

anti-quark

4 quark system

5 quark system

5 Quark Wilson Loop

4 Quark Wilson Loop

VNQ(r) = － lim ln<WNQ>T

1TT→∞

The Multi-Quark potentials can be obtained from the corresponding Multi-Quark Wilson Loops.

Okiharu, H.S. et al. PRL 94 (2005) 192001 Okiharu, H.S. et al. PRD72 (2005) 014505 We formulate Multi-Quark Wilson Loops.


VNQ(r)

N=4,5

quark

quark

quark

quark

quark

quark

anti-quark

anti-quark

anti-quark

4 quark system

5 quark system

For more than 200 different patterns of multi-quark configurations, we have accurately performed the first lattice QCD calculations for multi-quark potentials.

Partial lattice QCD data of Multi-quark potential



VNQ(r)

N=4,5

quark

quark

quark

quark

quark

quark

anti-quark

anti-quark

anti-quark

4 quark system

5 quark system

color flux tube





VNQ(r)

N=4,5

quark

quark

quark

quark

quark

quark

anti-quark

anti-quark

anti-quark

4 quark system

5 quark system

color flux tube





VNQ(r) = ∑ + σLming2

4πTi

aTja

|ri - rj|i<j

NLmin ： total length of string linking the N valence quarks


Linear potentialbased on string picture

VNQ(r)

N=4,5

quark

quark

quark

quark

quark

quark

anti-quark

anti-quark

anti-quark

4 quark system

5 quark system

color flux tube


2d

h

Okiharu, H.S. et al. PRD72 (2005) 014505

Summary of the First Part ~Static Potentials~

We have performed the first accurate Lattice QCD studies for static multi-quark (3Q, 4Q, 5Q) potentials.

The multi-quark potential is well described by OGE Coulomb+ String-picture Linear Confinement Potential.

VNQ(r) = ∑ + σLming2

4πTi

aTja

|ri - rj|i<j

NLmin ： total length of string linking the N valence quarks


Linear potentialbased on string picture

We have found the Universality of Quark Confinement Force (String Tension) in hadrons: σQQ =σ3Q =σ4Q =σ5Q-

Heavy-Heavy-Light Quark Potential and Light-quark Effects to Inter-two-quark Interaction

in Baryons~SU(3) Lattice QCD and Analytical Model

Calculation~So far, we have obtained the definite conclusions for the static inter-quark potentials in QCD.

However, in the real world, the quark mass is finite and quarks are moving inside hadrons.

Here, we investigate the effect of the quark motion to the inter-two-quark interaction in baryons.

To this end, we study the idealized situation of heavy-heavy-light quark systems where two heavy quarks can be treated as static quarks.

So far, we have obtained the definite conclusions for the static inter-quark potentials in QCD.

However, in the real world, the quark mass is finite and quarks are moving inside hadrons.

Here, we investigate the effect of the quark motion to the inter-two-quark interaction in baryons.

To this end, we study the idealized situation of heavy-heavy-light quark systems (QQq systems) where two heavy quarks can be treated as static quarks.

This situation physically corresponds to “doubly charmed baryon” as .

static quarks

idealize

Doubly charmed baryonIn 2002, the first doubly charmed baryon wasexperimentally observed at SELEX, Fermilab.

M. Mattson et al., Phys. Rev. Lett. 89, 112001 (2002).A. Ocherashvili et al., Phys. Rev. Lett. B628, 18 (2005).

Theoretical calculations for the doubly charmed baryons

Mass : 3519 ± 1 MeV

R. Lewis et al., Phys. Rev. D 64, 094509 (2001).N. Mathur et al., Phys. Rev. D 66, 014502 (2002).

Lattice QCD :

A. D. Rujula et al., Phys. Rev. D 12, 147 (1975).Potential model :

Doubly charmed baryon (experiment)

The first experimental observation at SELEX, Fermilab600 GeV/c charged hyperon beam

M. Mattson et al., Phys. Rev. Lett. 89, 112001 (2002).

2Mass 3519 ± 1 MeV/c

W. M. Yao et al., J. Phys. G 33, 1 (2006).

W. M. Yao et al., J. Phys. G 33, 1 (2006).

The situation is idealized as

Two heavy quarks (Q) → Two static quarks (MQ→∞)One light quark (q) → finite-mass quark (Mq: various value)

The QQq potential VQQq(R) is defined as the energy of QQq systems in terms of the inter-heavy-quark distance R.

Heavy-heavy-light quark (QQq) potentialHeavy-heavy-light quark (QQq) potential

R R

We calculate the QQq potential VQQq(R) in Lattice QCDand also in a non-relativistic potential model.

One light quark is moving around Two Static Quarks

QQq potential in Lattice QCD

The QQq Wilson loop is defined as

The QQq potential is obtained as

: light-quark propagator

cf) 3Q Wilson loop for static 3Q potential

• Standard plaquette gauge action• β= 6.0 (lattice spacing a = 0.10 fm) 　• 164 isotropic lattice• Quenched calculation• O(a)-improved clover fermion action• κ=0.1200, 0.1300, 0.1340, 0.1380• Wall-source wall-sink propagator• Calculated on NEC-SX8R at RCNP, Osaka Univ.

Lattice QCD Simulation Conditions

Constituent quark mass:

Mq = mρ/2

Lattice QCD Result for QQq potential

The QQq potential VQQq(R) can be well fitted by a Coulomb + linear type potential

The “effective” string tension between two heavy quarks is about 20% reduced.

cf.) Static 3Q in lattice QCD [Takahashi, H.S. et al., PRD 65, 114509 (2002).]

The Coulomb coefficient Aeff

is almost the same.

QQq Potential in the Potential ModelWe start from Non-relativistic Hamiltonian for heavy-heavy-light quark (QQq) system

using the static 3Q potential from the lattice QCD result

: the color flux-tube length

We calculate the light-quark wave function from this Hamiltonian, using the energy variational calculation in discretized space.

Note that, in the static limit of two heavy quarks, spin-dependent interactions proportional to 1/MQ disappear.

[Takahashi, H.S. et al., Phys. Rev. D 65, 114509 (2002).]

Confinement potential in Baryons

Z

ρ

Z

ρ

Confinement potential ~

If all angles of the 3Q triangle < 120°

If an angle of the 3Q triangle > 120°

Takahashi, H.S. et al., Phys. Rev. D 65, 114509 (2002).

Renormalization-group inspired variational method

To get the light-quark wave function,we perform the energy variational calculation in discretized space.

Here, we use the renormalized-group (RG) inspired variational method:

Z

ρ

Starting from a coarse lattice, we proceed the variational calculation to the finer mesh lattice iteratively.

Light-quark spatial distribution

The light-quark probability |Ψq|2 around the Two Static Quarks.Brighter region denotes higher probability of the light-quark.

Lattice QCD Result

The QQq potential is well fitted by

Potential Model Result

We find again about 20% reduction of the “effective” string tension between two heavy quarks.

cf.) Static 3Q in lattice QCD [Takahashi, H.S. et al., PRD 65, 114509 (2002).]

“Effective” String Tension between two heavy quarks

One reason for the reduction of the effective string tension between two heavy quarks is the geometrical difference between the inter-heavy-quark distance R and the flux-tube length Lmin.

Effective string tension

String tension

R

Lmin

R

L min

cf.) Meson

Summary and Concluding Remarks

• We have perform the first study for the QQq potential in SU(3) lattice QCD with O(a)-improved clover action and in the non-relativistic potential model.

• We have used a Renormalization Group (RG) inspired variational calculation for the calculation of light-quark wave function in the potential model.

• The effective string tension between two heavy quarks is significantly reduced in comparison with the ordinary string tension of the static 3Q case.

Light-quark Effects for Effective Reduction of Confining Force between Two Quarks in

BaryonsH. Suganuma, A. Yamamoto, H. Iida (Kyoto Univ.)

1. Heavy-heavy-light quark potential in SU(3) lattice QCD A. Yamamoto, H. Suganuma, H. Iida, arXiv:0708.3610 [hep-lat] Abstract: We perform the first study for the heavy-heavy-light quark (QQq) potential in SU(3) lattice QCD at the quenched level. We calculate the energy of QQq systems as the function of the distance R between the two heavy quarks, and find that the QQq potential VQQq(R) is well described with a Coulomb plus linear potential form. Compared with the static three-quark case, the effective string tension between the heavy quarks is significantly reduced by the light-quark effect.2. Light-quark effects on the inter-quark potential in baryons A. Yamamoto, H. Suganuma, arXiv:0709.0171 [hep-ph]

Abstract: We also study the QQq system in a non-relativistic potential model with the static three-quark potential which is obtained by lattice QCD. Using a renormalization-group-inspired variational method in discretized space, we calculate the ground-state energy of QQq systems and the light-quark spatial distribution. We find that the effective string tension between the heavy quarks is reduced compared to the static three-quark case. We conjecture that the effective reduction of the inter-two-quark confining force is induced by the remaining “3rd”quark in light-quark baryons.

Inter-Quark Potentials in Baryons and Multi-Quark Systems in QCD H. Suganuma, A. Yamamoto, H. Iida, N. Sakumichi (Kyoto Univ.) with T.T.Takahashi (Kyoto.

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