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 , RCNP Osaka Contents 1. Three-Quark Potential in SU(3) lattice QCD 2. Multi-Quark Potential in SU(3) lattice QCD 3. 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 QCD In 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 QCD M.Creutz (1979,80) quarkanti-quark Wilson loop t T r The quark-antiquark potential can be obtained from the Wilson Loop.

Quark-antiquark static potential in Lattice QCD r 0 =0.5fm:unit quarkanti-quark V(r) = － lim ln T 1T1T T→∞ Wilson loop t T r The quark-antiquark potential can be obtained from the Wilson Loop. M.Creutz (1979,80) Summarized lattice QCD data G.S.Bali (2001) Takahashi, H.S. et al. (2002) JLQCD (2003)

Quark-antiquark static potential in Lattice QCD V(r) = － +σr g 2 3π 1r1r r 0 =0.5fm:unit quarkanti-quark M.Creutz (1979,80) Summarized lattice QCD data G.S.Bali (2001) Takahashi, H.S. et al. (2002) JLQCD (2003) 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. Quark-antiquark static potential in Lattice QCD V(r) = － +σr g 2 3π 1r1r r 0 =0.5fm:unit g g quarkanti-quark － quarkanti-quark M.Creutz (1979,80) Summarized lattice QCD data G.S.Bali (2001) Takahashi, H.S. et al. (2002) JLQCD (2003)

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. Quark-antiquark static potential in Lattice QCD V(r) = － +σr g 2 3π 1r1r r 0 =0.5fm:unit － quarkanti-quark M.Creutz (1979,80) Summarized lattice QCD data G.S.Bali (2001) Takahashi, H.S. et al. (2002) JLQCD (2003)

Quark-antiquark static potential in Lattice QCD V(r) = － +σr g 2 3π 1r1r r 0 =0.5fm:unit quark anti-quark g g quarkanti-quark One-dimensional squeezing of color flux between q and q － M.Creutz (1979,80) Summarized lattice QCD data G.S.Bali (2001) Takahashi, H.S. et al. (2002) JLQCD (2003) quarkanti-quark Wilson loop t T r

Baryonic Three-Quark Potential in Lattice QCD 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.

PR This is the cover of a recent textbook written by Hosaka and Toki.

This is a nice textbook for the introduction to quark-hadron physics

PR This is the cover of a recent textbook written by Hosaka and Toki. 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) (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) (4 pages). “Detailed Analysis of the Gluonic Excitation in the 3Q System in Lattice QCD” T.T. Takahashi and H. Suganuma Physical Review D70 (2004) (13 pages). “Gluonic Excitation of the Three-Quark System in SU(3) Lattice QCD” T.T. Takahashi and H. Suganuma Physical Review Letters 90 (2003) (4 pages). “Detailed Analysis of the Three Quark Potential in SU(3) Lattice QCD” T.T. Takahashi, H. Suganuma et al. Physical Review D65 (2002) (19 pages). “Three-Quark Potential in SU(3) Lattice QCD” T.T. Takahashi, H. Suganuma et al. Physical Review Letters 86 (2001)

V 3Q (r) = － lim ln T 1T1T T→∞ t

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

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

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

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

i j k ( i, j, k ) characterize the shape of the 3Q triangle. -ml 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.

L min ： total length of string linking three valence quarks quark color electric flux

Baryonic Three-Quark Potential in Lattice QCD quark What Shape of Color Flux? Confining Force? Before our study, there was almost No lattice QCD study for the Three-Quark Potential Takahashi, H.S. et al. PRL 86 (2001) 18 Takahashi, H.S. et al. PRD65 (2002) Takahashi, H.S. PRL 90 (2003) Takahashi, H.S. PRD70 (2004) Okiharu, H.S. et al. PRD72 (2005)

Baryonic Three-Quark Potential in Lattice QCD V 3Q (r) conf quark color electric flux Takahashi, H.S. et al. PRL 86 (2001) 18 Takahashi, H.S. et al. PRD65 (2002) Takahashi, H.S. PRL 90 (2003) Takahashi, H.S. PRD70 (2004) Okiharu, H.S. et al. PRD72 (2005)

Baryonic Three-Quark Potential in Lattice QCD V 3Q (r) V 3Q (r) = ∑ + σL min g 2 4π T i a T j a |r i - r j | i<j 3 L min ： total length of string linking three valence quarks One-Gluon-Exchange Coulomb potential Linear potential based on string picture quark conf color electric flux Takahashi, H.S. et al. PRL 86 (2001) 18 Takahashi, H.S. et al. PRD65 (2002) Takahashi, H.S. PRL 90 (2003) Takahashi, H.S. PRD70 (2004) Okiharu, H.S. et al. PRD72 (2005)

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 3 rd edition of “Lattice Gauge Theories”, which is one of the most popular lattice QCD text books.

PR This is the cover of a recent textbook written by Hosaka and Toki. ?

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

PR This is the cover of a recent textbook written by Hosaka and Toki. 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.

PR This is the cover of the proceedings of Confinement Conference.

Multi-Quark Hadrons and Multi-Quark Potentials 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 release The charged charmonium Z + (4430) is a manifest Tetra-Quark hadron composed by ccud. --

Multi-Quark Hadrons and Multi-Quark Potentials 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. 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) Okiharu, H.S. et al. PRD72 (2005) quark anti-quark 4 quark system What Shape of Color Flux? Confining Force? ?

First Lattice QCD Study for Static Quark Potential in Multi-Quark System quark anti-quark 5 quark system What Shape of Color Flux? Confining Force? ? Okiharu, H.S. et al. PRL 94 (2005) Okiharu, H.S. et al. PRD72 (2005)

First Lattice QCD Study for Static Quark Potential in Multi-Quark System quark anti-quark 4 quark system 5 quark system 5 Quark Wilson Loop 4 Quark Wilson Loop V NQ (r) = － lim ln T 1T1T T→∞ The Multi-Quark potentials can be obtained from the corresponding Multi-Quark Wilson Loops. Okiharu, H.S. et al. PRL 94 (2005) Okiharu, H.S. et al. PRD72 (2005) We formulate Multi-Quark Wilson Loops.

First Lattice QCD Study for Static Quark Potential in Multi-Quark System V NQ (r) N=4,5 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 Okiharu, H.S. et al. PRL 94 (2005) Okiharu, H.S. et al. PRD72 (2005)

First Lattice QCD Study for Static Quark Potential in Multi-Quark System V NQ (r) N=4,5 quark anti-quark 4 quark system 5 quark system color flux tube 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 Okiharu, H.S. et al. PRL 94 (2005) Okiharu, H.S. et al. PRD72 (2005)

First Lattice QCD Study for Static Quark Potential in Multi-Quark System V NQ (r) N=4,5 quark anti-quark 4 quark system 5 quark system color flux tube 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 Okiharu, H.S. et al. PRL 94 (2005) Okiharu, H.S. et al. PRD72 (2005)

First Lattice QCD Study for Static Quark Potential in Multi-Quark System V NQ (r) = ∑ + σL min g 2 4π T i a T j a |r i - r j | i<j N L min ： total length of string linking the N valence quarks One-Gluon-Exchange Coulomb potential Linear potential based on string picture V NQ (r) N=4,5 quark anti-quark 4 quark system 5 quark system color flux tube Okiharu, H.S. et al. PRL 94 (2005) Okiharu, H.S. et al. PRD72 (2005)

2d h Okiharu, H.S. et al. PRD72 (2005)

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. V NQ (r) = ∑ + σL min g 2 4π T i a T j a |r i - r j | i<j N L min ： total length of string linking the N valence quarks One-Gluon-Exchange Coulomb potential Linear potential based on string picture We have found the Universality of Quark Confinement Force (String Tension) in hadrons: σ QQ = σ 3Q = σ 4Q = σ 5Q -

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 baryon In 2002, the first doubly charmed baryon was experimentally observed at SELEX, Fermilab. M. Mattson et al., Phys. Rev. Lett. 89, (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, (2001). N. Mathur et al., Phys. Rev. D 66, (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, Fermilab 600 GeV/c charged hyperon beam M. Mattson et al., Phys. Rev. Lett. 89, (2002). 2 Mass 3519 ± 1 MeV/c

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

The situation is idealized as Two heavy quarks (Q) → Two static quarks (M Q →∞) One light quark (q) → finite-mass quark (M q : various value) The QQq potential V QQq (R) is defined as the energy of QQq systems in terms of the inter-heavy-quark distance R. Heavy-heavy-light quark (QQq) potential RR We calculate the QQq potential V QQq (R) in Lattice QCD and 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) 16 4 isotropic lattice Quenched calculation O(a)-improved clover fermion action κ=0.1200, , , Wall-source wall-sink propagator Calculated on NEC-SX8R at RCNP, Osaka Univ. Lattice QCD Simulation Conditions Constituent quark mass: M q = m ρ /2

Lattice QCD Result for QQq potential The QQq potential V QQq (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, (2002).] The Coulomb coefficient A eff is almost the same.

QQq Potential in the Potential Model We 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/M Q disappear. [Takahashi, H.S. et al., Phys. Rev. D 65, (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, (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 |Ψ q | 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, (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 L min. Effective string tension String tension R L min R L 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 Baryons H. 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: [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 V QQq (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: [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.