The static three-quark (3Q) potential is studied in SU(3) lattice QCD with 12 3 × 24 and β = 5.7 at the quenched level. From the 3Q Wilson loop, 3Q ground-state potential V3Q is extracted using the smearing technique for ground-state enhancement. With accuracy better than a few %, V3Q is well described by a sum of a constant, the two-body Coulomb term and the three-body linear confinement term σ3QLmin, with Lmin the minimal value of total length of color flux tubes linking the three quarks. Comparing with the Q-Q potential, we find a universal feature of the string tension, σ3Q ≃ σ QQ , and the OGE result for Coulomb coefficients, A3Q ≃ 1 2In usual, the three-body force is regarded as a residual interaction in most fields of physics. In QCD, however, the three-body force among three quarks is expected to be a "primary" force reflecting the SU(3) c gauge symmetry. Indeed, the three quark (3Q) potential [1,2,3] is directly responsible to the structure and properties of baryons [4], similar to the relevant role of the Q-Q potential upon meson properties [5]. In contrast with a number of studies on the Q-Q potential using lattice QCD [6,7], there were only a few lattice QCD studies for the 3Q potential done mainly more than 13 years ago [8,9,10,11]. (In Ref.[11], the author only showed a preliminary result on the equilateral-triangle case without enough analyses.) In Refs. [8,10,11], the 3Q potential seemed to be expressed by a sum of twobody potentials, which supports the ∆-type flux tube picture [12]. On the other hand, Ref. [9] seemed to support the Y-type flux-tube picture [2,4] rather than the ∆-type one. These controversial results may be due to the difficulty of the accurate measurement of the 3Q ground-state potential in lattice QCD. For instance, in Refs. [8,10], the authors did not use the smearing for ground-state enhancement, and therefore their results may include serious contamination from the excited-state component.The 3Q static potential can be measured with the 3Q Wilson loop, where the 3Q gauge-invariant state is generated at t = 0 and is annihilated at t = T , as shown in Fig.1. Here, the three quarks are spatially fixed in R 3 for 0 < t < T . The 3Q Wilson loop W 3Q is defined in a gauge-invariant manner aswithHere, P denotes the path-ordered product along the path denoted by Γ k in Fig.1. Similar to the derivation of the Q-Q potential from the Wilson loop, the 3Q potential V 3Q is obtained asPhysically, the true ground state of the 3Q system, which is of interest here, is expected to be expressed by the flux tubes instead of the strings, and the 3Q state which is expressed by the three strings generally includes many excited-state components such as flux-tube vibrational modes. Of course, if the large T limit can be taken, the ground-state potential would be obtained. However, W 3Q decreases exponentially with T , and then the practical measurement of W 3Q becomes quite severe for large T in lattice QCD simulations. Therefore, for the accurate measurement of the 3Q ground-state potentia...

The static three-quark (3Q) potential is studied in detail using SU(3) lattice QCD with 12 3 × 24 at β = 5.7 and 16 3 × 32 at β = 5.8, 6.0 at the quenched level. For more than 300 different patterns of the 3Q systems, we perform the accurate measurement of the 3Q Wilson loop with the smearing method, which reduces excited-state contaminations, and present the lattice QCD data of the 3Q ground-state potential V3Q. We perform the detailed fit analysis on V3Q in terms of the Y-ansatz both with the continuum Coulomb potential and with the lattice Coulomb potential, and find that the lattice QCD data of the 3Q potential V3Q are well reproduced within a few % deviation by the sum of a constant, the two-body Coulomb term and the three-body linear confinement term σ3QLmin, with Lmin the minimal value of the total length of color flux tubes linking the three quarks. From the comparison with the Q-Q potential, we find a universality of the string tension as σ3Q ≃ σ QQ and the one-gluon-exchange result for the Coulomb coefficients as A3Q ≃ 1 2 A QQ . We investigate also the several fit analyses with the various ansätze: the Y-ansatz with the Yukawa potential, the ∆-ansatz and a more general ansatz including the Y and the ∆ ansätze in some limits. All these fit analyses support the Y-ansatz on the confinement part in the 3Q potential V3Q, although V3Q seems to be approximated by the ∆-ansatz with σ∆ ≃ 0.53σ.Even at present, the arguments on the 3Q potential seem to be rather controversial. In Refs. [24,26,27,28,29], the 3Q potential seemed to be expressed by a sum of two-body potentials, which supports the ∆-type flux tube picture [30]. On the other hand, Ref. [3,25,31,32,33] seemed to support the Y-type flux-tube picture [10,20] rather than the ∆-type one. These controversial results may be due to the difficulty of the accurate measurement of the 3Q ground-state potential in lattice QCD. For instance, in Refs.[24,26], the authors did not use the smearing for ground-state enhancement, and therefore their results may include serious contamination from the excited-state component. In Refs.[27,28,29], the author showed a preliminary result only on the equilateral-triangle case without the fit analysis.In this paper, for more than 300 different patterns of the 3Q system, we perform the accurate measurement of the static 3Q potential in SU(3) lattice QCD at β=5.7, 5.8 and 6.0, using the smearing method to remove the excited-state contaminations and to obtain the true ground-state potential. The contents are organized as follows. In Section 2, we make a brief theoretical consideration on the form of the inter-quark potential based on QCD. In Section 3, we explain the method of the lattice QCD measurement of the 3Q potential, referring the importance of the smearing technique and its physical meaning. In Section 4, we present the lattice QCD data of the 3Q potential, which are accurately measured from the smeared 3Q Wilson loop in a model-independent manner. In Section 5, we perform the fit analysis of the lattice data with the Y-...

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The static penta-quark (5Q) potential V5Q is studied in SU(3) lattice QCD with 16 3 × 32 and β=6.0 at the quenched level. From the 5Q Wilson loop, V5Q is calculated in a gauge-invariant manner, with the smearing method to enhance the ground-state component. V5Q is well described by the OGE plus multi-Y Ansatz: a sum of the OGE Coulomb term and the multi-Y-type linear term proportional to the minimal total length of the flux-tube linking the five quarks. Comparing with QQ and 3Q potentials, we find a universality of the string tension, σ QQ ≃ σ3Q ≃ σ5Q, and the OGE result for Coulomb coefficients.

We study the nucleon-nucleon interaction in the isoscalar-scalar channel using the chiral unitary approach. The t-matrix of the pion-pion scattering in this channel is summed up to all orders using the B-S equation. We find that the calculated results at long distances are close to those of the σ exchange interaction. In addition, there appears a shorter range repulsion in this channel. §1. IntroductionThe intermediate range attraction in the NN interaction has been traditionally described by the σ exchange in the meson exchange picture. 1) It has also been noticed 2) -4) that box diagrams with two pion exchange and intermediate ∆ excitation lead to an intermediate range attraction. A weakened σ exchange together with these box diagrams has also been used to describe the intermediate range NN attraction. 1)With the success of chiral perturbation theory (χP T ) in the meson-meson and meson-baryon sectors, 5) attempts to extend these ideas to the NN sector have been pursued. 6) -8) In a recent paper, 9) following the line of Ref. 8), the peripheral NN partial waves are studied within a chiral scheme, using the meson-meson and the meson-baryon interaction Lagrangians. With this input, together with more conventional processes with ∆ box diagrams, ρ exchange and other meson exchange, a good reproduction of the NN data for L > 2 partial waves is obtained.A striking feature of Ref. 9) is that the exchange of two interacting pions in the scalar-isoscalar channel (the σ channel) leads to a repulsion (although weak), instead of the commonly accepted attraction from σ exchange. Simultaneously, from the box diagrams with intermediate ∆, an attraction is obtained with the range and strength of the standard σ exchange of the boson exchange models. Two basic approximations lead to these results. First, the ππ isoscalar interaction is used only to lowest order in χP T . Second, no form factors are considered for the ∆ box diagrams.In the present work we wish to reconsider this idea and go further by treating the exchange of two interacting pions in the isoscalar channel in a nonperturbative way. The approach followed here for the ππ interaction produces a σ pole in the complex plane in the physical region s > 4m 2 π . Then the analytical extrapolation of the model is used in order to find the strength of the isoscalar exchange interaction

We perform the detailed study of the tetraquark (4Q) potential V4Q for various QQ-Q Q systems in SU(3) lattice QCD with β = 6.0 and 16 3 × 32 at the quenched level. For about 200 different patterns of 4Q systems, V4Q is extracted from the 4Q Wilson loop in 300 gauge configurations, with the smearing method to enhance the ground-state component. We calculate V4Q for planar, twisted, asymmetric, and large-size 4Q configurations, respectively. Here, the calculation for large-size 4Q configurations is done by identifying 16 2 × 32 as the spatial size and 16 as the temporal one, and the long-distance confinement force is particularly analyzed in terms of the flux-tube picture. When QQ and Q Q are well separated, V4Q is found to be expressed as the sum of the one-gluon-exchange Coulomb term and multi-Y type linear term based on the flux-tube picture. When the nearest quark and antiquark pair is spatially close, the system is described as a "two-meson" state. We observe a flux-tube recombination called as "flip-flop" between the connected 4Q state and the "two-meson" state around the level-crossing point. This leads to infrared screening of the long-range color forces between (anti)quarks belonging to different mesons, and results in the absence of the color van der Waals force between two mesons.

We present the first study of the gluonic excitation in the three-quark (3Q) system in SU(3) lattice QCD with β=5.8 and 16 3 × 32 at the quenched level. For the spatiallyfixed 3Q system, we measure the gluonic excited-state potential, which corresponds to the flux-tube vibrational energy in the flux-tube picture. The lowest gluonic-excitation energy in the 3Q system is found to be about 1GeV in the hadronic scale. This large gluonic-excitation energy is expected to bring about the success of the simple quark model without gluonic modes.The low-lying hadrons are well categorized with the simple quark model, which has only quark degrees of freedom. The success of the quark model implies the absence of the gluonic excitation in the low-lying hadron spectra, which seems rather mysterious. To understand the success of the quark model based on QCD, it is important to investigate the gluonic excitation mode.Theoretically, the gluonic excitation is expected to appear as the vibrational mode of the squeezed color-flux-tube, which is formed due to color confinement [1,2]. Experimentally, this type of the gluonic excitation is closely related to the hybrid mesons or the hybrid baryons, which consist of qqG or qqqG, respectively, in the valence picture. In particular, the several states with the exotic quantum number such as.. cannot be constructed with the simple quark picture [3], and therefore they are now investigated with much attention from both theoretical and experimental viewpoints [4].For the Q-Q system corresponding to mesons, there are several lattice QCD studies for both the ground-state and the excited-state potentials. The lattice QCD study indicates that the Q-Q ground-state potential takes the form of V g.s. QQ (r) = −A 1 r + σr, as the function of the inter-quark distance r, with A ≃ 0.27 and σ ≃0.89GeV/fm besides a irrelevant constant [5,6]. On the other hand, the lowest excited-state potential for the spatially-fixed Q-Q system can be fitted as V e.s.QQ (r) = √ b 0 + b 1 r + b 2 r 2 + c 0 [7]. Here, the quantum number of the first-excited state is L = 1 for the magnitude of the projection of the total angular