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PoS(LATTICE 2007)319 http://pos.sissa.it/ ∗ [email protected] [email protected] [email protected] [email protected] Speaker. We investigate the Gribov-Zwanziger scenario in the Coulombtice gauge gauge using simulation. a The SU(3) dressing quenched function lat- ofThis the result ghost is diverges expected in from the the infrared fact limit. near that the the vanishing Faddeev-Popov eigenvalue eigenvalue density compared gets to concentrated transversal that propagator in is the not abelian observed up gauge toThe theory. the instantaneous largest The part lattice of turnover volume explored the of in time-time the this componentthe study. of color-Coulomb the potential gluon in propagator which the corresponds continuumthermore, to limit we diverges observe stronger that than the ghost thegluon propagator simple propagator show pole. do good not scaling Fur- show while scaling both in components the of considered the range of the lattice spacing. ∗ Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. c

The XXV International Symposium on LatticeJuly 30 Theory - August 4 2007 Regensburg, Germany Hiroshi Toki Research Center for Nuclear Physics, OsakaIbarakisi, University Osaka 567-0044, Japan E-mail: Takuya Saito Integrated Information Center, Kochi University Kochi, 780-8520, Japan E-mail: Atsushi Nakamura Research Institute for Information Science andHigashi-Hiroshima Education, 739-8521, Hiroshima Japan University E-mail: Yoshiyuki Nakagawa Color confinement and the Faddeev-Popov ghosts in Coulomb gauge QCD Research Center for Nuclear Physics, OsakaIbarakisi, University Osaka 567-0044, Japan E-mail: PoS(LATTICE 2007)319 (1.3) (1.4) (1.5) (1.1) (1.2) . ) the F-P ghost oper- Yoshiyuki Nakagawa y − M x ]. Recently lattice QCD ( . 5 ) P t , z ~ , ( is the color charge density ) + z . ,~ b i 4 ab y ~ ρ y ∂ ρ ]) ) tr , − tr A c i [ 4 A A , 1 x ] , the Faddeev-Popov (F-P) ghosts ; − ( z 2 abc ,~ δ , M y a ) ~ ) 1 ( y g f ~ ρ 2 ab ∇ − − + x V 2 ~ i − tr ( ) , ∂ c t i ](

, V ab E A 2 i y ~ [ tr ( δ 2 , 1 B a b i ) = − − ρ + A y z M 2 3 = ) − abc d tr i x ab i ( E g f Z D ( ) = ( i 44 y tr  = ∂ 3 D A x d a 2 ; − 3 g z ρ d ,~ Z = y ~ = 1 2 Z ( i ab 2 1 ) ab + y M ( V 4 = A ) H x ( 4 A ]. Furthermore, it was demonstrated that the contribution of the ghost loop h 3 2 g the kernel of the instantaneous interaction ], 4 In the Coulomb gauge, the importance of the F-P ghosts and the confinement scenario are more About ten years ago, the Landau gauge study of the gluon and the ghost Dyson-Schwinger To understand the mechanism of color confinement is a challenging issue in particle and nu- V whose is calledator the color-Coulomb potential, and From the partition function withcomponent the of Coulomb the gauge gluon Hamiltonian, propagator one composedpart can by [ evaluate the the instantaneous time-time part and the non-instantaneous It was shown bybound Zwanziger for the that static the potential andpotential instantaneous is the that color-Coulomb necessary the condition potential color-Coulomb for potential provides the is static also an a potential confining upper being potential a [ confining The first term represents themagnetic energy fields. of the transverse The componentsoriginates second from of the part, the energy color-electric of the and the instantaneous color- longitudinal color-electric interaction fields. energy between color charges, and simulation showed that the instantaneous color-Coulomband it potential is rises stronger linearly than at the large static potential; distances this is an expected result from the Zwanziger’s inequality transparent. The Coulomb gauge is atwo physical parts, gauge and the Hamiltonian can be decomposed into in the gluon DSE iscontribution crucial is for numerically the small IRtheory in is suppression the dominated of perturbation by the the theory, unphysical gluon the ghost propagator. IR degrees of dynamics Although freedom, of the i.e., ’IR the ghost ghost Yang-Mills dominance’. equations (DSEs) revealed thatthan the ghost the propagator simple diverges pole,zero in and the momentum the infrared [ gluon (IR) propagator limit is stronger suppressed in the IR region and vanishes at Color confinement and the Faddeev-Popov ghosts in Coulomb gauge QCD 1. Introduction clear physics. There have beenZwanziger several and scenarios the proposed Kugo-Ojima forplay color confinement a confinement. scenarios significant role [ In in the the Gribov- mechanism of color confinement. PoS(LATTICE 2007)319 (2.2) (2.3) (2.1) . i Yoshiyuki Nakagawa ˆ i − x , y δ ) ˆ i of the F-P ghost operator is the lattice 3-volume) and − 1 x − 3 ]. On the boundary of the Gri- ( i V 8 M y , U ]. x b δ T 13  i a y ) ,~ ˆ i T . ab x ~ 2 ) ]) − | p x − A ~ 2 [ | ˆ ( i i ( 1 p ~ + − G x U as , y sparse matrix ( Z i M δ ( 3 U ) ab ) + h x V 3 x δ ( ( 8 i i U U × ) = a ) = y 3 ~ p T } ~ ]. Such an enhancement has been observed by the recent V − b ( b 9 x T ~ T ab ( , 2 G a ab − T G { h . Since the ghost propagator is diagonal in color space, the ghost Tr a i A Re i ]. Accordingly, the ghost propagator becomes more singular than the free ∑ 11 , = 10 ab xy ]. The vanishing of the equal-time transverse gluon propagator would imply that M 12 [ | p ~ | is the dressing function of the ghost propagator. The non-trivial momentum dependence p G / Z ]. Therefore, in the Coulomb gauge, the color confinement is attributed to the instantaneous The ghost propagator is defined as the vacuum expectation value of the inverse of the F-P ghost In the Gribov-Zwanziger confinement scenario, the strong confining feature of the instan- On a lattice, the F-P operator is an 8 In this study we perform a lattice investigation of the IR behavior of the ghost and the gluon The analysis of the ghost DSE in the Coulomb gauge has revealed that the equal-time trans- 7 , 6 depends on the gauge fields propagator in the momentum space can be written as In contrast with the abelian , the Green’s function operator, expressed in terms of SU(3) spatial link variables Here of the ghost propagator due to the quantum correction is encoded in the dressing function. 2. Lattice observables in order to clarify thedone confinement by mechanism Voigt et. in al the has Coulomb also gauge. been A reported similar in work this conference [ have infinite effective mass inons. the Coulomb By gauge, contrast, which indicates the the time-timethe confinement simple component of pole of glu- in the the gluonis IR propagator responsible limit would for since diverges the it stronger color contains than confinement. the instantaneous color-Coulomb potential, which propagator and the IR enhancementthe of color-Coulomb the potential ghost which propagator is leads responsible to for the the color long-range confinement. interaction of verse gluon propagator vanishes at zero momentumthan if 1 the ghost dressing function diverges stronger taneous interaction originates fromdiscussed, the the strong Coulomb long gauge range doesrestricted not correlation to fix the of a Gribov F-P region gauge ghosts. where completely, the and F-P As the operator gauge Gribov is configurations positive are [ Color confinement and the Faddeev-Popov ghosts in Coulomb gauge QCD [ color-Coulomb interaction. bov region, so-called the Gribovwas argued horizon, by the Zwanziger that lowest entropy eigenvalue favorseigenvalue gauge of distribution configurations the of near F-P the the Gribov operator F-P horizonpared vanishes. operator and to the gets that It in concentrated the near abelian gauge thelattice theory simulations vanishing [ [ eigenvalue com- PoS(LATTICE 2007)319 (2.5) (2.6) (2.4) 3 [GeV]. = and the lattice | µ p ~ | lattice gauge sim- Yoshiyuki Nakagawa ) 3 ( SU 10 , ) y ~ . ) − | x p =6.00 =6.10 =6.00 =6.00 =5.80 =5.90 ~ ~ | ( β β β β β β ( , , , , , , 4 4 4 4 4 4 tr µν ab , 24 24 12 18 24 24 D ) D |  p ~ = 2 j | ( 2 i p p ~ ) i in the continuum limit. p 44 ~ y p ~ ( Z V b ν − ab A 4 p [GeV] ) δ i j x δ ~ (  µ a 1 ) = A p ab ~ h ( δ ab 44 ) = D ) = y ~ p ~ ( − x ~ ab i j ( D 1 2

µν ab

2.5 1.5 G

(|p|) D Z 1. ]) = the ghost dressing function is plotted as a function of the magnitude of the three 1 GeV [ The ghost dressing function in the confinement phase. As a condition, we set 3 . In this study, the lattice propagators are renormalized multiplicatively at = Λ | The lattice propagators depends on the magnitude of the three momentum We calculate the ghost propagator and the gluon propagator by the The equal-time gluon propagator is given by In Fig. p ~ | ( G cutoff and the transversal equal-time gluon propagator in the momentum space can be written as The instantaneous part of the time-time component of the gluon propagator is ulations in a .Monte Carlo The technique lattice with the configurationstive Wilson are method plaquette to generated . fix by a In the gauge. theseby heat-bath In simulations the order we conjugate to gradient adopt obtain method the the and ghost itera- be used propagator published plane we elsewhere. have wave inverted sources. the The F-P details matrix of the calculation3.1 will Ghost propagator momentum. Although the finite volume effect isgood seen scaling. at As small is lattice expected volume, the in propagator the shows Gribov-Zwanziger scenario, the ghost dressing function shows Figure 1: Z which corresponds to the color-Coulomb potential 3. Results and discussions Color confinement and the Faddeev-Popov ghosts in Coulomb gauge QCD PoS(LATTICE 2007)319 5 (3.1) =5.80 =5.80 =5.80 β β β , , , 4 4 4 18 24 32 Yoshiyuki Nakagawa p [GeV] 10 , =5.80 =5.90 =6.00 ) β β β , , , 1 4 4 4 ( 24 24 24 42 . . 1 gh 0 1 2 3 4

γ 5

0

15 ) 10 1 ] [GeV (p)

= D 2

c

80 to the IR power law ansatz -1 . tr 1 p ~ c 5 ( 5 = p [GeV] , 5 ) ) = β | 0, which indicates the confinement of gluons. 2 1 p ( ~ | = ( 29 . G p ~ 0 Z =5.80 =5.80 =5.80 β β β = , , , lattice at 4 4 4 4 18 24 32 gh γ

1 3 5 8 7 9 2 0 4 6

10 ] [GeV D

-1 tr p [GeV] The equal-time transversal gluon propagator at various lattice couplings. 244. The analysis of the ghost DSE in the Coulomb gauge has revealed that . ]. From our result of the ghost dressing function we expect that the equal-time 0 12 The equal-time transverse gluon propagator without (left) and with (right) the cone cut. = Figure 3: nd f / 2 0 1 2 3 4

5

χ 0

15 In order to explore the momentum dependence of the ghost dressing function in the IR region,

10 Figure 2: ] [GeV (p) D

-1 tr with the equal-time transversal gluon propagatorexceeds vanishes 0.25 at zero [ momentum if the ghost IR exponent We found the IR exponent transversal gluon propagator vanishes at Color confinement and the Faddeev-Popov ghosts in Coulomb gauge QCD a divergent behavior in theinstantaneous IR interaction. limit, which results in the confining feature ofwe the fitted data color-Coulomb below 1 [GeV] on 24 3.2 Equal-time transverse gluon propagator PoS(LATTICE 2007)319 . 2 ]. The turnover of Yoshiyuki Nakagawa 14 ]. 10 15 =5.90 =6.10 β=5.80 β β=6.00 β , , , , 4 4 4 4 24 24 24 24 ]. 13 6 p [GeV] 1

1 5 3 7 6 4 2 0 44

(p) Z shows the transversal gluon propagator at different lattice couplings. We observe that 3 The instantaneous part of the time-time gluon propagator at various lattice couplings. Both the . We see that the dressing function diverges in the IR region, and this is consistent with the 4 We investigated the ghost and the gluon propagators in the Coulomb gauge using the quenched The lattice simulation results for the transversal gluon propagator are presented in Fig. The dressing function of the instantaneous part of the time-time gluon propagator is illustrated fact that the color-Coulomb potentialshows obtained a confining by behavior. measuring Scaling theas is partial well not Polyakov seen as line for the correlator thepropagator instantaneous transversal was time-time also gluon observed gluon propagator. in propagator the SU(2) Non-scaling lattice gauge of simulations the [ instantaneous time-time gluon lattice simulations. We found thatan the IR ghost divergent dressing behavior function of the divergeswhich in F-P ghosts the confines leads infrared color to limit. the charges. Such strongthe long-range The vanishing instantaneous ghost interaction equal-time IR gluon exponentgluon propagator was propagator at found does vanishing to not momentum. show beBoth the turnover The about transversal up equal-time 0.29; and to transversal the this the time-timethe implies component largest considered of lattice range the volume gluon of propagators explored thereconsidered. do in lattice not this coupling show scaling study. constant, in and the renormalization procedure must be We find that the finite volume effect is removed by applying the cone cut [ 3.3 Instantaneous part of the time-time gluon propagator Figure 4: cylinder and the cone cut are applied. in Fig. 4. Summary and conclusion the transversal propagator does not showeffect. scaling This and has is also faced been observed with by the Voigt strong et finite al. lattice [ spacing Color confinement and the Faddeev-Popov ghosts in Coulomb gauge QCD the transversal gluon propagator isstudy. not Fig. observed up to the largest lattice volume explored in this PoS(LATTICE 2007)319 , 031501 (1998). , 338 (2007), 58 Yoshiyuki Nakagawa LAT2007 , 014508 (2007), [hep-lat/0702002]. D75 , 3591 (1997), [hep-ph/9705242]. 79 , 070 (2005), [hep-lat/0407032]. , 1 (1979). , 189 (2006), [hep-lat/0512042]. 05 66 , 014001 (2001). , 054005 (2005), [hep-ph/0504244]. 7 115 65 , 233 (1998), [hep-th/9802180]. D72 , 094503 (2003), [hep-lat/0302018]. 131 D67 , 102001 (2003), [hep-lat/0209105]. , 477 (1993). , 657 (1994). , 1 (1978). 90 B399 B412 B139 [arXiv:0709.4585 [hep-lat]]. The simulation was performed on SX-5 and SX-8(NEC) vector-parallel computers at the [1] D. Zwanziger, Nucl. Phys. [2] T. Kugo and I. Ojima, Prog.[3] Theor. Phys. L. Suppl. von Smekal, R. Alkofer and A. Hauck, Phys. Rev. Lett. [4] D. Zwanziger, Prog. Theor. Phys. Suppl. [5] D. Zwanziger, Phys. Rev. Lett. [6] J. Greensite and S. Olejnik, Phys.[7] Rev. A. Nakamura and T. Saito, Prog. Theor. Phys. [8] V. N. Gribov, Nucl. Phys. [9] D. Zwanziger, Nucl. Phys. Color confinement and the Faddeev-Popov ghosts in Coulomb gauge QCD Acknowledgements RCNP of Osaka University. We appreciate thetrators. warm One hospitality of and the support authors of (Y.N.) the acknowledgesfor the RCNP Short-term support adminis- Student from the Dispatch Osaka Program. University Scholarship References [10] J. Greensite, S. Olejnik and D.[11] Zwanziger, JHEP Y. Nakagawa, A. Nakamura, T. Saito and H. Toki, Phys. Rev. [12] C. S. Fischer and D. Zwanziger, Phys. Rev. [13] A. Voigt, E.-M. Ilgenfritz, M. Mueller-Preussker and A. Sternbeck, PoS [15] A. Cucchieri and D. Zwanziger, Phys. Rev. D [14] D. B. Leinweber, J. I. Skullerud, A. G. Williams and C. Parrinello, Phys. Rev. D