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Color confinement in Coulomb gauge QCD and color-dependent interactions

Color confinement in Coulomb gauge QCD and color-dependent interactions. Takuya Saito 斎藤卓也. Collaborators ： A.Nakamura （ Hiroshima ） ,H.Toki （ RCNP),Y.Nakagawa （ RCNP),D. Zwanziger (NY). 共同研究者：中村純（広大）、土岐博（ RCNP) 、中川義之（ RCNP) 、 D. Zwanziger (NY). Part1:

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Color confinement in Coulomb gauge QCD and color-dependent interactions

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  1. Color confinement in Coulomb gauge QCD and color-dependent interactions Takuya Saito 斎藤卓也 Collaborators：A.Nakamura（Hiroshima）,H.Toki（RCNP),Y.Nakagawa（RCNP),D. Zwanziger (NY) 共同研究者：中村純（広大）、土岐博（RCNP)、中川義之（RCNP)、D. Zwanziger (NY) 東京大学ハドロン研究室セミナー

  2. Part1: Study of color confinement scenario in Coulomb gauge: lattice calculation of color-Coulomb Instantaneous potential in color singlet channel~ Part2: Lattice study on color-dependent potentials of QCD; lattice study of the color 3* quark-quark potential, and 8 quark-antiquark, 6 qq potentials. 東京大学ハドロン研究室セミナー

  3. Study of color confinement scenario in Coulomb gauge~ lattice calculation of color-Coulomb instantaneous potential ~ Takuya Saito （RCNP at Osaka Univ.） • Collaborators： • Y. Nakagawa （RCNP at Osaka Univ.） • H. Toki （RCNP at Osaka Univ.) • Nakamura (RIISE at Hiroshima Univ.) • D. Zwanziger ( NY Univ.) 東京大学ハドロン研究室セミナー

  4. Contents Motivation Color confinement scenario in the Coulomb gauge QCD Method ( partial-length Polyakov line ) Numerical results（in the confinement and deconfinement phases） Summary 東京大学ハドロン研究室セミナー

  5. Confinement of the quarks and gluons in the hadron.One can not detect an isolated quark. However, the quarks and gluons give a good description for hadrons. • In QCD lattice simulation, the quark potential rises linearly for the large quark separation, implying the non-vanishing string tension. • However, there is a problem how QCD produces the confinement of the quarks and gluons. Confinement 東京大学ハドロン研究室セミナー

  6. There were several approaches and a lot of works to understand the confinement …. : Confinement • Dual superconductor scenario, centre vortex model, the infrared behavior of gluon propagators, etc. • Topological quantities in the QCD vacuum are important：magnetic monopole, instanton, centre vortex, etc. • A proper gauge fixing should be used. In this study, we focus the Coulomb gauge QCD, and we will investigate the confinement mechanism in Coulomb gauge by the lattice QCD simulation. 東京大学ハドロン研究室セミナー

  7. Confinement scenario of Coulomb gauge QCD （By Zwanziger） • Coulomb instantaneous potential in QCD • Difference between Wilson-loop and instantaneous potentials • FP-ghost operator and instantaneous potentials • Related topics for Coulomb gauge D. Zwanziger, PTP Suppl. No. 131, 233(1998); A.Cucchieri, D.Zwanziger, PRD65,014001,(2002). PRD65,014002,(2002) 東京大学ハドロン研究室セミナー

  8. Hamiltonian in the Coulomb gauge QCD Faddeev-Popov term in the Coulomb gauge QCD Time-time component of the gluon propagators. Coulomb gauge QCD retarded (vacuum polarization) part Instantaneous part 東京大学ハドロン研究室セミナー

  9. Color-Coulomb instantaneous part Important quantity in the Coulomb gauge confinement scenario • Vcoul(r) : Instantaneous part for the quark-antiquark potential.　（antiscreening effect）. We conjecture that this term produces the color confinement. • P(x,t) : Retarded (vacuum polarization), not instantaneous part　（screening effect）.This term contributes the pair quark creation if the dynamical quark is alive. 東京大学ハドロン研究室セミナー

  10. Quark Wilson loop potential and color-Coulomb instantaneous potential • Quark Wilson loop potentail, Vw ,should be distinguished from color-Coulomb instantaneous potentail Vc. • Color-Coulomb, Ｖｃ, is responsible for confinement. 東京大学ハドロン研究室セミナー

  11. Zwanziger’s inequality Zwanziger, PRL90, 102001 (2003) Here the physical potential corresponds to the Wilson loop potential. If the physical potential is confining, then the color-Coulomb potential is also confining. 東京大学ハドロン研究室セミナー

  12. Fadeev-Popov and instant. parts Instantaneous part is defined in terms of FP operator in QCD It is conjecturd by Gribov that the low-lying mode of eigenvalues of FP causes the singular behavior of the potential ( producing the string tension); namely, their low-lying mode is responsible for asthe color confinement. 東京大学ハドロン研究室セミナー

  13. Related refs. for the Coulomb gauge QCD (1) • Study of confinement by Gribov. NPB139,1 (1978) • Color-Coulomb instantaneous part is very important, which is advocated by Zwanziger, NPB518,237 (1998) • Study of the renormalization of the Coulomb gauge QCD, Baulieu, Zwanziger, NPB548,527(1998) • By the SU(2) lattice simulation, it is proved that the infrared part, D00(k=0), shows the large contributions, while the spatial part Dii (k=0) is suppressed.( Cucchieri, Zwanziger, PRD65,0142002,(2002) ) • There is an inequality, Vphys <=Vcoul, which is found by Zwanziger, PRL90, 102001 (2003) • The SU(2) lattice simulation shows that the instantaneous part is confining potential; namely it rises linearly at the large distances. ( Greensite, Olejnik, PRD67,094503(2003),PRD69,074506(2004). ) 東京大学ハドロン研究室セミナー

  14. Related refs. for the Coulomb gauge QCD (2) • The SU(3) lattice simulation shows that the instantaneous part is the confining linearly rising force, and in the deconfinement phase, the instantaneous potential is also a linearly rising potential, but the retarded part causes the QGP screening effect. ( Nakamura, Saito、PTP115(2006)189-200.) • Recently, in the QGP phase, we discussed the relation between the non-vanishing color-Coulomb string tension and the non-vanishing Wilson loop string tension in the spatial direction in terms of the magnetic scaling. ( Nakagawa, Nakamura, Saito, Toki, Zwanziger, hep-lat-0603010, PRD73(2006)094504) 東京大学ハドロン研究室セミナー

  15. Aim in this study • By the SU(3) lattice simulation, we study the behavior of the color-Coulomb instantaneous potential for large quark separations in the hadron ( confinement ) and QGP ( deconfinement ) phase. • We would like to study the scaling behavior of the color-Coulomb string tension obtained by the instantaneous part: • The asymptotic scaling in the confinement phase. • The magnetic scaling in the deconfinement, for the non-vanishing string tension. 東京大学ハドロン研究室セミナー

  16. Method • Quantizaion by lattice regularizaion • Gauge fixing on lattice gauge theory • Measurement ( partial-length polyakov loops ) 東京大学ハドロン研究室セミナー

  17. Lattice regularization • Lattice regularization • cut-off • link variable • Wilson action • Path-integral quantization 東京大学ハドロン研究室セミナー

  18. Lattice regularization • Expectation value and Monte Carlo method Expectation values we want Gauge configurations are generated by the probability After N times repeated, one can obtain physical quantities 東京大学ハドロン研究室セミナー

  19. Monte Carlo Steps Gauge rotation Wilson-Mandula Method PLB185,127(1987) Gauge fixing on a lattice • In general, a gauge fixing is not required in finite size lattices. • Iterative method to fix gauge confs. 東京大学ハドロン研究室セミナー

  20. Measurement In this study, the most important issue is to extract the instantaneous part from the gluon propagators. PRD67,094503(2003),PRD69,074506(2004). Partial-length Polyakov line • Here, V(R,0) corresponds to the instantaneous Vcoul(R). • V（R,1), V(R,2), ．．． are the vacuum ( retarded ) parts, which are not important now. 東京大学ハドロン研究室セミナー

  21. Simulation parameters One plaquette Wilson gauge action and quenched sim. Lattices at zero temp.：β=5.85-6.40, 184, 183x32, ３00 confs. Lattices at finite temp.: β=6.11~7.0, 243x6, 300 confs. A la Mandula-Oglive method for gauge fixing （maximization of ReTrU） Computer facilities : ＮＥＣ　ＳＸ５　of RCNP at Osaka Univ. 東京大学ハドロン研究室セミナー

  22. Numerical results:(1)for the confining phase 東京大学ハドロン研究室セミナー

  23. Color-Coulomb potential （confining phase） • V(R,0) is a linearly rising potential, i.e., confining potential. • The potentials including a retarded part approach the Wilson loop potential. • We can fit the data by the Coulomb plus linear terms. • Zwanziger’s inequality is satisfied. instantaneous retarded (vacuum) PTP115(2006)189-200 東京大学ハドロン研究室セミナー

  24. Scaling of Coulomb string tension Asymptotic scaling Beta function 東京大学ハドロン研究室セミナー

  25. Scaling of the color-Coulomb string tension • If the asymptotic scaling of QCD is satisfied enough, then we will find the following relation: • Color-Coulomb string tension scales monotonically as the lattice cutoff or the coupling constant. 東京大学ハドロン研究室セミナー

  26. Numerical results:(2)for the deconfining phase 東京大学ハドロン研究室セミナー

  27. Color-Coulomb potential（deconfining phase) : the typical behavior • Instantaneous part gives still the linearly confining potential. Very remarkable feature. • Color-Coulomb string tension is not an order parameter of QGP phase transition. • The potential with the (full) retarded part is the color-screened Yukawa-type potenial. PTP115(2006)189-200 東京大学ハドロン研究室セミナー

  28. Color-Coulomb potential（deconfining phase) : at higher temperature • Linearity of instantaneous part dose not vanish at high temperature. • Appearance of any non-perturbative mode !? • Instantaneous part , not having explicitly the time variable, may not be sensitive to time (temperature) variable. 東京大学ハドロン研究室セミナー

  29. Review of temp. dep. of the spatial string tension G.S. Bali, et. al, PRL71,3059(1993) Spatial Wilson loop gives the finite spatial string tension, which increases with the temperature. • This behavior is very similar to that of the instantaneous potential. • Spatial Wilson loop and instantaneous parts are independent on time ( temperature ) variable. • Their two spatial quantities will be described mainly by the spatial gluon prop. with the magnetic (pole) mass. 東京大学ハドロン研究室セミナー

  30. Temp. dep. of the spatial string tension G.S. Bali, et. al, PRL71,3059(1993) Spatial quantities at finite temperature are expected to be described by the magnetic scaling, which is believed to dominate the high temp. QCD. Usually, the following assumption is used, This assumption is good for the data over T/Tc=2. Here, let’s assume that the instantaneous part also satisfies the magnetic scaling. 東京大学ハドロン研究室セミナー

  31. Comparison with magnetic scaling • Color-Coulomb string tension can be described by the magnetic scaling. • However, the fitting by the electric scaling is not too bad, and in the temp. region, the coupling constant is still O(1). • In any cases, it is clear that there exist the color-Coulomb string tensions after the QGP phase transition, which are scaled with the temperature. log scale 東京大学ハドロン研究室セミナー

  32. T dep. of instantaneous string tension Fitting function 東京大学ハドロン研究室セミナー

  33. T dep. of instantaneous string tension • Two-parameter fit ( T/Tc=2-4 ) • Spatial Wilson loop; two-parameter fit, ( NPB469 1996 410-444 ) • Spatial gluon propagator ( PRD69,014506,2004 ) • If we use the electric scaling… ( T/Tc = 2-4 ) It may be less proper since leading order perturbation gives C=1. 東京大学ハドロン研究室セミナー

  34. Summary 東京大学ハドロン研究室セミナー

  35. Summary We have investigated the behavior of the color-Coulomb instantaneous potentials in the confinement/deconfinement phase. We discussed the asymptotic scaling of the color-Coulomb string tensions in the confinement phase, while in the deconfinement phase, the comparison with the magnetic scaling is made. Retarded (vacuum polarization) part of the gluon prop. is responsible for color-screening effect: it weakens the color-Coulomb string tension in the confinement phase, while in the deconfinement phase, it produces the screened potential. 東京大学ハドロン研究室セミナー

  36. In conclusion, it is clear that the color-Coulomb instantaneous potential is a source of color confinement; however, the color-Coulomb string tension is not an order parameter of the QGP phase transition. It might indicate the remnant of color confining force in the QGP phase. These are remarkable features of the Coulomb gauge QCD: In connection with the understanding with the Coulomb gauge Hamiltonian, the strongly interaction QGP system, etc. Summary 東京大学ハドロン研究室セミナー

  37. Future work Color-Coulomb instantaneous potential is very closely related to the singularity of Faddeev-Popov operator. This is Gribov conjecture (example) and we should the eigenvalue distribution of FP operator. Application to the phenomenology of the hadron or QGP systems. (although we have no idea yet.) Calculation of the color-dependent potential among two or three quarks potential. Investigate of the non-instantaneous vacuum polarization ( retarded ) parts. It may relate to the QGP phase transition, the chiral symmetry breaking, the pair quark creation, etc. 東京大学ハドロン研究室セミナー

  38. Lattice study oncolor-dependent potentials of QCD Takuya Saito in collaboration with A. Nakamura This presentation is based on PLB621(2005)171,PTP111(2004)733,PTP112(2004)183 and in collaboration with H. Toki and Y. Nakagawa 東京大学ハドロン研究室セミナー

  39. Contents QCD color quark potential Polyakov loop correlator Numerical results Summary 東京大学ハドロン研究室セミナー

  40. Introduction Color potentials in QCD 東京大学ハドロン研究室セミナー

  41. Quarks have 3 color degree of freedom and we have to consider several color potentials depending on each color channel. For example, in SU(3) color group Color potentials in QCD Forces among color sources are characterized in the quadratic Casimir Factor. Color-dep. forces are important for studies of multi-quark states, di-quark model, color-super conductor, etc. Here, we want to investigate those by lattice QCD simulation. 東京大学ハドロン研究室セミナー

  42. Quark-antiquark potential in color singlet channel. • Attractive. C=-4/3. Strongest force in two-quark potentials. • For understanding of the dynamics of color confinement and making a hadron state • Linearly rising behavior in the hadron phase. • Color-screened potentials in the QGP phase. • Widely studied by lattice QCD simulations. • But, the gauge invariant Wilson loop or Polyakov loop cannot distinguish between color-singlet and color octet channels ! Singlet potential 東京大学ハドロン研究室セミナー

  43. Antisymmetric potential • Quark-quark potential in color antisymmetric 3* • Attractive.C＝-2/3. • A diquark picture is very important under several situations: Multiquark system, highly correlated qq interaction ? Also very important in finite chemical system. ( although lattice simulations are not working now … ) • Behavior in the hadron and QGP phases ? • Linearly rising potential in the hadron phase？ • Screened potentials in the QGP phase ? • It has not been studied by lattice QCD simulation ! 東京大学ハドロン研究室セミナー

  44. Color-octet potential • Quark-antiquark in color octet 8 • Repulsive. C=1/6. Weakest force in two-quark pot. • Precise measurement of J/Ψphotoproduction: color-octet model (CLEO Collab. hep-ex/0407030, Cacciari and Kramer, PRL76,4128(1999)). • Multi-quark and hybrid hadrons: the description of the ccg system ( if a color octet pot. is attractive ? ). • For understanding of QGP • Not studied well by lattice QCD simulations. 東京大学ハドロン研究室セミナー

  45. Symmetric potential • Quark-quark potential in symmetric channel • Repulsive, C=１/3. • Multi-quark and hybrid hadrons • For understanding of QGP • Not studied by lattice QCD simulation at all. 東京大学ハドロン研究室セミナー

  46. Study of the color-dependent forces is very important in the hadron and QGP phases. But, now, there are few lattice studies. The Wilson loop calculation does not yield the color-dependent forces, because it, for example, mixes the contributions of 1 and 8. Our aim in this study Here, we use the correlator functions of the not-gauge invariant Polyakov loop with Coulomb gauge and investigate the long-distance behavior of the color-dependent potential by lattice QCD simulation. 東京大学ハドロン研究室セミナー

  47. Our aim in this study • Quark-antiquark：color-singlet, color-octet channel • Quark-quark：color-antisymmetric, color-symmetric • Check Casimir scalings for the string tension. • Behavior in finite temperature system ? 東京大学ハドロン研究室セミナー

  48. Polyakov loop correlators • Polyakov line • Polyakov line correlator • Potentials between two quarks • Partial-Polyakov line correlator 東京大学ハドロン研究室セミナー

  49. Polyakov line • Polyakov line • Order parameter in pure gauge theory ( McLerran, RMP58, 1021(1986) ) 東京大学ハドロン研究室セミナー

  50. Polyakov line correlator • Two-quark state at t=0 • Quark-antiquark potential 東京大学ハドロン研究室セミナー

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