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)

.

2.S (U x,μ)B x,μ=

1

2

,B x,μ

trlp.

(4)

The fundamental modular gauge (FMG)[3]is speci?ed by the global minimum of the minimizing function F U (g )along the gauge or-bits in either case,i.e.,

Λ={U |A =A (U ),F U (1)=Min g F U (g )},Λ??,where ?is Gribov region (local min-ima),and ?={U |??D (U )≥0,?A =0}.

3

2.Numerical simulation of lattice Landau

gauge QCD

2.1.Method of simulation

First we produce Monte Carlo(Boltzmann) samples of link variable con?guration according to Wilson’s plaquette action by using the heat-bath method.We use optimally tuned combined algorithm of Creutz’s and Kennedy-Pendleton’s for SU(2)case,and Cabbibo-Marinari pseudo-heat-bath method with use of the above SU(2) algorithm for SU(3)case.As for FMG?xing, the smearing gauge?xing works well for largeβon relatively small size lattice[8].However we found that its performance for SU(3),β=6.0, 164lattice is not perfect in comparison with our standard method for log U type de?nition,in which the third order perturbative treatment of the linear equation with respect to gauge?eld A,??D?=?A is performed.For U linear de?ni-tion,we use the standard overrelaxation method for both SU(2)and SU(3).Thus we know that our gauge?xing is not FMG?xng.The accuracy of?A(U)=0is10?4in the maximum norm in all cases.

Measurement of quantity in question,gluon propagator,ghost propagator,Kugo-Ojima pa-rameter,etc.,is performed with use of Landau gauge?xed copy,and averaged over Monte Carlo samples.

We de?ne the gluon dressing function Z A(q2) from the gluon propagator of SU(n)

Dμν(q)=

1

q2

)D A(q2),(5) as Z A(q2)=q2D A(q2).

The ghost propagator is de?ned as the Fourier transform of an expectation value of the inverse of Faddeev-Popov(FP)operator M=??D,

D ab G(x,y)= tr λa x|(M[U])?1|λb y ,(6) where the outmost ··· denotes average over samples U.

2.2.Numerical results of simulation

D A(q2)is de?ned in(5),and its dressing func-tion is given as Z A(q2)=q2D A(q2).Numerical

Figure1.The gluon dressing function as the function of the momentum q(GeV).β=6.4, 484(stars)and564(diamonds)in the log U de?ni-tion.The solid line is that of the MOM scheme. results are given in?gure.1.

The ghost propagator is de?ned in(6).It should be noted that the operator[M(U)]ab xy is a real symmetric operator when?A=0, and it transforms covariantly under global color gauge transformation g,i.e.,[M(U g)]ab= [g?{M(U)}g]ab.The conjugate gradient method is used for the evaluation the ghost propagator, and absence of color o?-diagonal components manifests itself,a signal of no color summetry violation in Landau gauge.The accuracy of the inversion is maintained less than5%in maximum norm in error check of the source term.

We measured the running coupling from the product of the gluon dressing function and the ghost dressing function squared.

αs(q2)=

g20

4

Figure2.The running couplingαs(q)as a func-tion of the logarithm of momentum log10q(GeV) of the log U,β=6.4,564lattice using the ghost propagator of the average(diamond).The result using the ghost propagator of I A copy(star)is also plotted for comparison.The DSE approach (long dashed line),the perturbative QCD+c/q2 (short dashed line)and the contour improved per-turbation method(dotted line)are also shown. for presenting the ambiguity due to the Gribov copy.

Kugo-Ojima color con?nement criterion is given with use of u ab(p2)de?ned in(1)with u ab(p2)=δab u(p2),as u(0)=?c;c=1→color con?nement.Zwanziger’s horizon condition [3]derived in the in?nite volume limit is written as follows;

x,y e?ip(x?y) tr λa?Dμ1

d pμpνp2 u ab,

where e= x,μtr(λa?S(U x,μ)λa) /{(n2?1)V},and the horizon condition reads lim

p→0

Gμμ(p)?e=0,and the l.h.s.of the con-dition is e d and dimension d=4,and it follows that h=0→horizon condition,and thus the horizon condition coincides with Kugo-Ojima criterion provided the covariant derivative approaches the naive continuum limit,i.e.,e/d=1.

Values of c,

e

5 Table1

The Kugo-Ojima parameter c in log U and U?linear version.β=6.0and6.4.

βc1e1/d h1

6.00.628(94)0.943(1)-0.32(9)

240.695(63)0.861(1)-0.17(6)

6.00.777(46)0.944(1)-0.16(5)

6.40.700(42)0.953(1)-0.25(4)

480.739(65)0.884(1)-0.15(7)

6.40.827(27)0.954(1)-0.12(3)

6

log U type has a maximum of about2.2,and that of the U?linear is slightly smaller since they are rescaled at the high lattice momentum region. The absolute value of the indexαG increases as the lattice size becomes large but since it is de-?ned at q～0.4GeV region,it is smaller than κ～0.5of the DSE which is de?ned at the0 momentum region by about factor2.

As is studied in SU(2)FMG data,the gluon propagator su?ers almost no Gribov noise,but magnitude of Kugo-Ojima parameter becomes smaller in FMG,and the ghost propagator ex-hibits less singular infrared behavior in FMG than that supposed su?ering noise.Accordingly the running coupling becomes smaller in FMG.All samples in SU(3)can not be supposed in FMG and supposed to su?er the Gribov noise,and it is expected that these qualitative aspects seen in SU(2)will re?ect in the infrared behavior of SU(3)QCD as well.The FMG is mathemat-ically well de?ned on lattice,and its existence can be proven.But rather?at valued feature of the FMG optimizing function with respect to physical quantities seem to keep annoying us,at least,technically,and this fact may suggest ne-cessity of some new formulation of infrared dy-namics of QCD.In the Langevin formulation of Landau gauge QCD,Zwanziger conjectured that the path integral over the FM region will become equivalent to that over the Gribov region in the continuum[13].If his conjecture is correct,then in order for the gauge dependent quantity,e.g., the optimizing function to have the same expecta-tion values on both regions,the situation that the probability density may be concentrated in some common local region becomes favorable one.The proximity of the FM region and the boundary of the Gribov region in SU(2)in84,124and164lat-tices withβ=0,0.8,1.6and2.7was studied in [14].The tendency that the smallest eigenvalue of the Faddeev-Popov matrix of the FMG and that of the1st copy come closer asβand lattice size become larger was observed,although as re-marked in[14]the physical volume ofβ=2.7, 164lattice is small and not close to the contin-uum limit.In this respect,further study of lattice size andβdependence of Gribov noise of various quantities will become an important problem in future.

Our observation of exceptional con?gurations in SU(3),β=6.4,564may be considered as a signal of the tendency of the probabililty concen-tration to the Gribov boundary[3,11].

The con?nement scenario was recently re-viewed in the framework of the renormalization group equation and dispersion relation[15].It is shown that the gluon dressing function satis-?es the superconvergence relation,and the gluon propagator does not necessarily vanish as Gri-bov and Zwanziger conjectured,but it should be ?nite.The multiplicative renormalizable DSE approach[5]predicts that the exponentκ=0.5 and the infrared?xed pointα0=2.6and our lattice data are consistent with this prediction. Acknowledgments:

We are grateful to Daniel Zwanziger for enlight-ning discussion.This work is supported by the KEK supercomputing project No.03-94,and No. 04-106.

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