We study the QCD color interactions between static two heavy quarks at zero
temperature in a quenched SU(3) lattice gauge simulation: in addition to the
standard singlet $q\bar{q}$ potentials, we calculate octet $q\bar{q}$
potentials, symmetric and antisymmetric $qq$ potentials. It is shown that the
antisymmetric $qq$ channel behaves as a linearly rising potential at large
quark separations. We further find that the $q\bar{q}$ octet and $qq$ symmetric
channels have the complex dependence on the distance; at short distances they
are repulsive forces, while at large distances, they show linearly rising
feature. Ratio of string tensions between $q\bar{q}$ singlet and $qq$
antisymmetric potentials is described in terms of the Casimir factor

Nakamura, A.

Saito, T.

Physics Letters B

English

arXiv

AbstractWe study the QCD color interactions between static two heavy quarks at zero temperature in a quenched SU(3) lattice gauge simulation: in addition to the standard singlet qq¯ potentials, we calculate octet qq¯ potentials, symmetric and antisymmetric qq potentials. It is shown that the antisymmetric qq channel behaves as a linearly rising potential at large quark separations. We further find that the qq¯ octet and qq symmetric channels have the complex dependence on the distance; at short distances they are repulsive forces, while at large distances, they show linearly rising feature. Ratio of string tensions between qq¯ singlet and qq antisymmetric potentials is described in terms of the Casimir factor

Elsevier - Publisher Connector 

QCD color interactions between two quarks 

Physics Letters B 621 (2005) 171–175www.elsevier.com/locate/physletbQCD color interactions between two quarksA. Nakamura, T. SaitoResearch Institute for Information Science and Education, Hiroshima University, Higashi-Hiroshima 739-8521, JapanReceived 7 May 2005; accepted 19 June 2005Available online 28 June 2005Editor: H. GeorgiAbstractWe study the QCD color interactions between static two heavy quarks at zero temperature in a quenched SU(3) lattice gaugesimulation: in addition to the standard singlet qq¯ potentials, we calculate octet qq¯ potentials, symmetric and antisymmetric qqpotentials. It is shown that the antisymmetric qq channel behaves as a linearly rising potential at large quark separations. Wefurther find that the qq¯ octet and qq symmetric channels have the complex dependence on the distance; at short distances theyare repulsive forces, while at large distances, they show linearly rising feature. Ratio of string tensions between qq¯ singlet andqq antisymmetric potentials is described in terms of the Casimir factor. 2005 Elsevier B.V.PACS: 12.38.Aw; 12.39.Mk; 12.38.Gc; 11.15.HaKeywords: Lattice QCD; Color octet channel; Color antisymmetric channel; DiquarkOpen access under CC BY license.1. IntroductionNonperturbative study of color QCD forces be-tween static quarks is important for understandingquark confinement and hadron phenomenology. Thebehavior of a heavy-quark potential that is defined bythe closed gauge-invariant Wilson loop, has been stud-ied extensively in lattice simulations [1]; the heavy-quark potential is a linearly rising potential, which canbe explained with the Coulomb term and the linearE-mail address: tsaito@rcnp.osaka-u.ac.jp (T. Saito).0370-2693  2005 Elsevier B.V.doi:10.1016/j.physletb.2005.06.053Open access under CC BY license.term with the string tension. In most previous stud-ies of heavy-quark potentials, a color-singlet channel,which yields a physical potential, has been investi-gated. There are, however, several other color chan-nels between two quarks. According to the SU(3)color decomposition for a quark–antiquark sector qq¯ ,3 ⊗ 3¯ = 1 ⊕ 8. The octet channel is significant forthe extensive analysis of J/ψ photoproduction [2,3].Furthermore, we have 3 ⊗ 3 = 3¯ ⊕ 6 for a quark–quark sector qq , and in particular, the antisymmet-ric qq channel plays an essential role in the phe-nomenology of penta-quark hadrons [4–6], in whichthe existence of a highly correlated diquark is as-172 A. Nakamura, T. Saito / Physics Letters B 621 (2005) 171–175sumed [5]. Since the usual Wilson loop cannot givethe color-decomposed potentials separately, the latticestudy along this line was considered as very diffi-cult.In order to understand quark–gluon plasma (QGP)physics [7], the finite-temperature behavior of theQCD color forces between two static heavy quarksis studied in lattice simulations. At finite tempera-ture, one expects that the linearly confining colorforce is weakened due to the screening effect in QGP.Quarks confined in hadrons are expected to movefreely in the deconfinement phase after the QGPphase transition. This can be understood in the non-perturbative behavior of the screened heavy-quarkpotentials defined by a Polyakov line correlator, theform of which is reduced to a Yukawa-type poten-tial with screening masses rather than the linearlyrising potential. We therefore may expect the massshift and/or suppression of heavy quarkoniums as anindicator of the QGP phase transition [8,9]. How-ever, recent heavy-ion collision experiments and thelattice QCD simulations [10,11] have indicated thatthis picture is too simple and QGP is a more com-plex system. Thus, in recent finite-temperature latticesimulations, the singlet and octet qq¯ potentials andthe symmetric and antisymmetric qq potentials havebeen investigated [12–15]. This is a progress com-paring with previous lattice QCD studies, where thescreened heavy-quark potential had focused on only acolor-averaged channel. On the other hand, the non-perturbative behavior of the SU(3) color-decomposedchannels at zero temperature has not been investi-gated.Recently, using SU(2) lattice gauge simulation,Greensite, Olejník and Zwanziger studied the colorconfinement mechanism at zero temperature and foundthat the Coulomb singlet heavy-quark potential inCoulomb gauge represents a strong linearly rising po-tential [16,17], and this was later verified in SU(3)lattice gauge simulation [18]. The Coulomb singletheavy-quark potential is calculated from partial-lengthPolyakov lines (PPL) in Coulomb gauge, not fromthe closed gauge-invariant Wilson loop. PPL is aPolyakov line with a restricted temporal extension[16,17,19]. Thus, the combinations of the PPL corre-lators may enable us to carry out lattice calculationsin the color-decomposed channels at zero tempera-ture.In this Letter, we study the behavior of the color-decomposed qq¯ and qq potentials at zero temperaturein the quenched SU(3) lattice gauge simulation. Ap-plying the PPL correlator method [16,17] to the color-decomposed potentials, we carry out the SU(3) latticecalculation of the octet, symmetric and antisymmet-ric potentials. We will present the first nonperturbativestudy of the antisymmetric qq potential at zero tem-perature.2. Partial-length Polyakov lineIn this section, we give color decomposed correla-tors between two static heavy quarks and summarizehow to fix the gauge on the lattice.A partial-length Polyakov line (PPL) can be definedas [16,17](1)L(x,T ) =T∏t=1U0(x, t), T = 1,2, . . . ,Nt .Here U0(x, t) = exp(iagA0(x, t)) is a SU(3) link vari-able in the temporal direction and a, g, A0(x, t) andNt represent the lattice cutoff, the gauge coupling, thetime component of gauge potential and the temporal-lattice size. A PPL correlator in the color-singlet chan-nel is given by(2)G1(R,T ) = 13〈Tr[L(R,T )L†(0, T )]〉,where R stands for |x|. From Eq. (2) one evaluate acolor-singlet potential on the lattice(3)V (R,T ) = log[G1(R,T )G1(R,T + a)].For the smallest temporal lattice extension, i.e., T = 0,we define(4)V (R,0) = − log[G1(R,1)].Greensite et al. argued that V (R,0) in Coulombgauge corresponds to a color-Coulomb potential[16,17]. The V (R,T ) at T → ∞ corresponds to aphysical potential, which is usually calculated fromthe Wilson loops at T → ∞. These two potentials areexpected to satisfy Zwanziger’s inequality, Vphys(R)Vcoul(R) [20].Now we extent this PPL correlator method tothe other SU(3) color decomposed potentials [12].A. Nakamura, T. Saito / Physics Letters B 621 (2005) 171–175 173A color-octet correlator for qq¯ is given byG8(R,T ) = 18〈TrL(R,T )TrL†(0, T )〉(5)− 124〈TrL(R,T )L†(0, T )〉,and qq correlators in the symmetric and antisymmetricchannels are described asG6(R,T ) = 34〈TrL(R,T )TrL(0, T )〉(6)+ 34〈TrL(R,T )L(0, T )〉,G3¯(R,T ) =32〈TrL(R,T )TrL(0, T )〉(7)− 32〈TrL(R,T )L(0, T )〉.In the same way as described in Eqs. (3) and (4), wewill obtain the color-decomposed potentials in eachcolor channel using the above correlators. The four po-tentials between two quarks are classified in terms ofthe Casimir factor on color SU(3) group in the funda-mental representation: C1qq¯ = − 43 for a singlet, C8qq¯ =16 for an octet, C6qq = 13 for symmetric, C 3¯qq = − 23 forantisymmetric; i.e., the qq¯ singlet and qq antisym-metric channels yield an attractive force, whereas theqq¯ octet and qq symmetric channels give a repulsiveforce.Since the color-decomposed potentials defined bythe PPL correlators do not have gauge-invariant forms,we must fix the gauge. We use the Coulomb gauge re-alized on the lattice as(8)Max∑x3∑i=1Re TrU†i (x, t),by repeating the following gauge rotations:(9)Ui(x, t) → Uωi (x, t) = ω†(x, t)Ui(x, t)ω(x + iˆ, t),where ω ∈ SU(3) is a gauge rotation matrix andUi(x, t) are spatial lattice link variables. Thus each lat-tice configuration can be gauge fixed iteratively [21].The Coulomb gauge fixing does not fix a gaugecompletely, and one can still perform a time-dependentgauge rotation on the Coulomb-gauge fixed links,Ui(x, t) → ω†(t)Ui(x, t)ω(t),(10)U (x, t) → ω†(t)U (x, t)ω(t + 1).0 0Thus, TrLTrL† and TrLL constructed by PPL, arenot invariant under the transformation of Eq. (10).Therefore, when performing numerical simulations forthe octet and two qq correlators including TrLTrL†and TrLL, we additionally implement a global tempo-ral-gauge fixing on the Coulomb-gauge fixed configu-ration as(11)Max 1V∑x,tRe TrU†0 (x, t) under Eq. (10),where V = NxNyNz is a spatial lattice volume andNx,y,z is the spatial lattice size in each direction. Notethat this gauge fixing does not affect intrinsic Coulombgauge features.3. Simulation resultsWe carry out SU(3) lattice gauge simulations inquench approximation to calculate the color decom-posed PPL correlators. The lattice configurations aregenerated by the heat-bath Monte Carlo techniquewith a plaquette Wilson gauge action, and to fix agauge we adopt the iterative method [21].Numerical results of the color decomposed poten-tials with T = 2 are displayed in Fig. 1. This sim-ulation is done on the 243 × 32 lattice at β = 5.90(a ∼ 0.124 fm) and the 200 configurations measuredby every 100 steps are used here. The singlet qq¯Fig. 1. Singlet and octet potentials for qq¯ sector and symmetric andantisymmetric potentials for qq sector. The solid curves representthe fitted results for the two attractive channels. The Rˆ with physicaldimension is set as Rˆ = aR where the lattice cutoff a is approxi-mately 0.124 fm for this lattice simulation.174 A. Nakamura, T. Saito / Physics Letters B 621 (2005) 171–175and the antisymmetric qq potentials rise linearly atlarge distances; i.e., they are an attractive confiningpotential. The octet qq¯ and the symmetric qq chan-nels show repulsive forces at short distances, while atlarge distances they have the linearly rising attractiveforce rather than the large-distance repulsive force.This nonperturbative behavior in the adjoint channelis comparable with the numerical result in 3d SU(2)gauge theory [22,23].In order to analyze the singlet qq¯ and the antisym-metric qq potentials, we employ the following fittingfunction,(12)V (R,T ) = C + KR + A/R, A = −π/12,where C is a constant and K corresponds to thestring tension. This fitting works well: approximatelyχ2/ndf ∼ O(1) for the fitting range R = 2–6. The re-sults are represented by solid lines in Fig. 1.We obtain the string tension K3¯ in the antisymmet-ric channel. The singlet-string tension K1 approachesasymptotically the Wilson-loop string tension [16,18],but the T dependence of K3¯ is unknown. Here, we in-vestigate the ratio of the string tensions obtained fromthe singlet qq¯ and the antisymmetric qq potentials; ifthe string tension itself is universal, then the differ-ence between K1 and K3¯ can be understood in termsof the color SU(3) Casimir factor; that is, K1/K3¯ →C1qq¯/C3¯qq = (−4/3)/(−2/3) = 2. Those ratios calcu-lated at β = 5.85–6.00 are summarized in Fig. 2. Thedependence of the ratio on β and T is weak and theyFig. 2. Ratios of string tensions in the singlet and antisymmetricchannels.are consistent with the expected values. We concludethat the relation between the singlet qq¯ and the anti-symmetric qq potentials is described in terms of thecolor SU(3) Casimir in nonperturbative regions.Since the repulsive channels have the complex de-pendence in the distance, we do not carry out a fittinganalysis for them.4. SummaryWe have nonperturbatively studied the zero-tem-perature behavior of the color decomposed poten-tials between static two heavy quarks in the quenchedSU(3) lattice gauge simulation with the partial-lengthPolyakov line correlators.We show that the qq potential in the antisymmetricchannel rises linearly at large distances and is fitted bythe function that includes the Coulomb and the string-tension linear terms. Moreover, the string tensions ob-tained from qq¯ singlet and qq antisymmetric poten-tials seem to be understood in terms of the Casimir; inother words, the string tension itself is universal. As aresult, not only the qq¯ singlet channel but also the qqantisymmetric channel is found to be strongly corre-lated in nonperturbative regions.The octet qq¯ and symmetric qq potentials show thecomplex behavior as a function of the distance in con-trast to the case of the attractive channels: the qq¯ octetand qq symmetric channels yield a repulsive force atshort distances, while at large distances it seems to bereduced to a linear potential. The larger lattice simula-tion therefore may be required to deal with the long-range features in the repulsive channels.In the case of color nonsinglet channels, it is espe-cially important to investigate the volume size effect,because the color nonsinglet potentials may includedivergent parts in the infinite volume limit.Our numerical simulation has been concentrated onthe two quarks system. With the PPL correlator wemay deal with multiquark potentials on the lattice. Thediscovery of the penta-quark hadron [4] promotes thenonperturbative lattice research about the multiquarkstate [6,24–26]. The SU(3) lattice simulation of tetra-and penta-quark potentials also has been carried out[24–26]. The PPL correlator enables us to constructthem without any assumption to make tentative formsA. Nakamura, T. Saito / Physics Letters B 621 (2005) 171–175 175of the multiquark potential, although the gauge-fixingprocedure is indispensable.The color decomposed potentials at zero temper-ature are computed in the quenched lattice simula-tion, and especially the behavior of the antisymmet-ric qq potential should be also investigated in a dy-namical quark simulation. The PPL correlator (or theusual Polyakov line correlator) is applicable to thedynamical-quark lattice simulation.AcknowledgementsWe would like to thank D. Zwanziger for very help-ful discussions. We are also grateful to H. Toki andV. Dmitrašinovic´ for useful comments. The simulationwas performed on SX-5(NEC) vector-parallel com-puter at the RCNP of Osaka University. We appre-ciate the warm hospitality and support of the RCNPadministrators. This work is supported by Grants-in-Aid for Scientific Research from Monbu-Kagaku-sho(Nos. 11440080, 12554008 and 13135216).References[1] G.S. Bali, Phys. Rep. 343 (2001) 1, hep-ph/0001312.[2] E. Braaten, S. Fleming, Phys. Rev. Lett. 74 (1995) 3327;E. Braaten, Y. Chen, Phys. Rev. Lett. 76 (1996) 730;M. Cacciari, M. Krämer, Phys. Rev. Lett. 76 (1996) 4128.[3] CLEO Collaboration, R.A. Briere, et al., Phys. Rev. D 70(2004) 072001, hep-ex/0407030.[4] LEPS Collaboration, T. Nakano, et al., Phys. Rev. Lett. 91(2003) 012002.[5] R.L. Jaffe, F. Wilczek, Phys. Rev. Lett. 91 (2003) 232003, hep-ph/0307341.[6] S. Sasaki, Phys. Rev. Lett. 93 (2004) 152001, hep-lat/0310014;S. Sasaki, hep-lat/0410016.[7] D.J. Gross, R.D. Pisarski, L.G. Yaffe, Rev. Mod. Phys. 53(1981) 43.[8] T. Hashimoto, O. Miyamura, K. Hirose, T. Kanki, Phys. 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QCD color interactions between two quarks

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