Numerical Study of Lattice Landau Gauge QCD and the Gribov C

a r X i v :h e p -l a t /0408001v 1 1 A u g 2004Numerical Study of Lattice Landau Gauge QCD and the Gribov Copy Problem Hideo Nakajima ?Department of Information Science,Utsunomiya University,321-8585Japan Sadataka Furui ?School of Science and Engineering,Teikyo University,320-8551Japan The infrared properties of lattice Landau gauge QCD of SU(3)are studied by measuring gluon propagator,ghost propagator,QCD running coupling and Kugo-Ojima parameter of β=6.0,164,244,324and β=6.4,324,484,564lattices.By the larger lattice measurements,we observe that the runnning coupling measured by the product of the gluon dressing function and the ghost dressing function squared rescaled to the perturbative QCD results near the highest lattice momentum has the maximum of about 2.2at around q =0.5GeV/c,and behaves either approaching constant or even decreasing as q approaches zero.The magnitude of the Kugo-Ojima parameter is getting larger but staying around ?0.83in contrast to the expected value ?1in the continuum theory.We observe,however,there is an exceptional sample which has larger magnitude of the Kugo-Ojima parameter and stronger infrared singularity of the ghost propagator.The re?ection positivity of the 1-d Fourier transform of the gluon propagator of the exceptional sample is manifestly violated.Gribov noise problem was studied by performing the fundamental modular gauge (FMG)?xing with use of the parallel tempering method of β=2.2,164SU(2)con?gurations.Findings are that the gluon propagator almost does not su?er noises,but the Kugo-Ojima parameter and the ghost propagator in the FMG becomes ～5%less in the infrared region than those su?ering noises.It is expected that these qualitative aspects seen in SU(2)will re?ect in the infrared properties of SU(3)QCD as well.1.Introduction

One of our basic motivations in the present study is veri?cation of the color con?nement mechanism in the Landau gauge.Two decades ago,Kugo and Ojima proposed a criterion for the color con?nement in Landau gauge QCD using the Becchi-Rouet-Stora-Tyutin(BRST)in-variance of continuum theory [1].Gribov pointed out that the Landau gauge can not be uniquely ?xed,that is,the Gribov copy problem,and ar-gued that the unique choice of the gauge copy could be a cause of the color con?nement [2].Later Zwanziger developed extensively the lat-tice Landau gauge formulation [3]in view of the Gribov copy problem.Kugo and Ojima started from naive Faddeev-Popov Lagrangian obviously ignoring the Gribov copy problem,and gave the color con?nement criterion with use of the follow-q 2)u ab (q 2)=1??D [A ν,λb ] xy ,(1)where lattice simulation counterpart is utilized.They claim that su?cient condition of the color con?nement is that u (0)=?1with u ab (0)=δab u (0).Kugo showed that 1+u (0)=Z 1?Z 3,(2)where Z 3is the gluon wave function renormaliza-tion factor,Z 1is the gluon vertex renormalization factor,and ?Z

3is the ghost wave function renor-malization factor,respectively.In the continuum theory ?Z

1is a constant in perturbation theory 1

2

and is set to be 1.On the lattice,it is not evident that it remains 1when strong non-perturbative e?ects are present.In a recent SU(2)lattice sim-ulation with several values of β,?niteness of ?Z

1seems to be con?rmed,but its value di?er from 1.

The same equality,the ?rst one of equations (2),was derived by

Zwanziger with use of his ”horizon condition”about the same time [3].It is to be noted that arguments of both Kugo and Zwanziger are perturbative ones in that they used diagramatic expansion,and the equation (2)is of continuum theory or continuum limit.

The non-perturbative color con?nement mech-anism was studied with the Dyson-Schwinger ap-proach [4,5]and lattice simulations [6,7,8,9,10].Both types of studies are complementary in that Dyson-Schwinger approach needs ansatz for trun-cation of interaction kernels and lattice simula-tion is hard to draw conclusions of continuum limit although the calculation is one from the ?rst principle.

We measured in SU(3)lattice Landau gauge simulation with use of two options of gauge ?eld de?nition (log U ,U linear;see below),gluon propagator,ghost propagator,QCD run-ning coupling and Kugo-Ojima parameter of β=6.0,164,244,324and β=6.4,324,484,564lattices.The QCD running coupling αs =g 2/4πcan be measured in terms of gluon dress-ing functiuon Z (q 2)and ghost dressing function G (q 2),as renormalization group invariant quatity g 2G (q 2)2Z (q 2).Infrared features of g 2is not known,however,and there remains a problem of checking the Gribov noise e?ect among those quantities,since there exist no practical algo-rithms available so far for ?xing fundamaental modular gauge (see below).In our 564simula-tion,we encountered a copy of an exceptional con?guration yielding extraordinarily large Kugo-Ojima marameter c =?u (0),and studied its fea-ture in some more detail.We made another copy by adjusting controlling parameter in gauge ?x-ing algorithm,and measured copywise 1-d FT of the gluon propagator,and found violation of re-?ection positivity in both cases.

In order to study the Gribov copy problem,we made use of parallel tempering [9]with 24replicas

to ?x fundamental modular gauge in SU(2),β=2.2,164lattice,and obtained qualitatively similar result as Cucchieri[12].

1.1.The lattice Landau gauge

We adopt two types of the gauge ?eld de?ni-tions:

1.log U type:U x,μ=e A x,μ,A ?x,μ=?A x,μ,

2.U linear type:A x,μ=

1

3

Re tr U g

x,μ ,respectively.Under in?nitesimal gauge transfor-mation g ?1δg =?,its variation reads for either de?ntion as

?F U (g )=?2 ?A g |? + ?|??D (U g )|? +···,where the covariant derivativative D μ(U )for two options reads commonly as

D μ(U x,μ)φ=S (U x,μ)?μφ+[A x,μ,ˉφ

]where ?μφ=φ(x +μ)?φ(x ),and ˉφ

=φ(x +μ)+φ(x )th(x/2)

.

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