Recent topics of infrared effective lattice QCD

a r X i v :h e p -l a t /9909104v 1 13 S e p 1999

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KANAZAWA 99-18

August 1999

Recent topics of infrared e?ective lattice QCD

?

Tsuneo Suzuki a ,Shun-ichi Kitahara b ,Fumiyoshi Shoji c ,Atsushi Nakamura c ,Kentarou Yamagishi a ,Shoichi Ito a ,Tomohiro Tsunemi a ,Shouji Fujimoto a ,Seikou Kato a and Hiroaki Kodama a

a Department of Physics,Kanazawa University,Kanazawa 920-1192,Japan b

Jumonji University,Niiza,Saitama 352-8510,Japan

c

Research Institute for Information Science and Education,Hiroshima University,Higashi-Hiroshima 739-8521,Japan

Three topics concerning infrared e?ective lattice QCD are discussed.(1)Perfect lattice action of infrared SU (3)QCD and perfect operators for the static potential are analytically given when we assume two-point monopole interactions alone.The assumption seems to be justi?ed from numerical analyses of pure SU (3)QCD in maximally abelian gauge.(2)Gauge invariance of monopole dominance can be proved theoretically if the gauge invariance of abelian dominance is proved.The gauge invariance of monopole condensation leads us to con?nement of abelian neutral but color octet states after abelian projection.(3)A stochastic gauge ?xing method is developed to study the gauge dependence of the Abelian projection,which interpolates between the maximally abelian (MA)gauge and no gauge ?xing.Abelian dominance for the heavy quark potential holds even in the gauge which is far from Maximally Abelian one.

1.SU3infrared e?ective monopole action Abelian dominance and monopole dominance in MA gauge suggest the existence of an e?ec-tive monopole action also in SU (3)QCD.There are two independent (three with one constraint 3

i =1k i

μ(s )=0)currents in the case of SU (3).Applying the same inverse Monte-Carlo method developed in the case of SU (2)QCD[1,2],we have derived the infrared e?ective monopole action

starting from the vacuum ensemble {k i

μ(s )}de-?ned from the thermalized abelian link ?elds after the MA abelian projection[3].E?ective monopole actions can be derived similarly for the blocked monopole currents K αμ(s )= n ?1i,j,m =0k αμ(ns +(n ?1)?μ+i ?ν+j ?ρ+m ?σ)[4].This corresponds to the block-spin transformation of the monopole cur-rents on the dual lattice.The form of the action is restricted to 27two-point interactions up to 3na (β)distance between monopole currents and the most leading 4and 6point interactions.The lattice size is 484for β=5.6～6.4.The results are as follow:

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selves to quadratic monopole interactions alone, the block-spin transformation can be done analyt-ically as done in SU(2)QCD[5].We can obtain also the perfect operator to evaluate the static quark-antiquark potential V(Ib,Jb,Kb)on the

coarse b lattice.

In the framework of the string model,the quan-

tum e?ects are seen to be small in the infrared re-

gion and the static potential is evaluated by the

classical contributions alone:

W(C) cl=exp{?2

2?μαβγ?Sβγ(s′+?μ)

and Sβγ(s)is the source term corresponding to

the Wilson loop.We get the rotational in-

variance V(Ib,Ib,0)/V(Ib,0,0)=

√

4

(δB,(?D)?1δB)

e1

4

(J,(?D)?1J) B=0

1)Electric-electric current J?J interactions(with

no monopole k)come from the exchange of

regular photons and have no line singularity

leading to a linear potential.Hence the lin-

ear potential of abelian Wilson loops is due to

the monopole contribution alone.Monopole

dominance is proved from abelian domi-

nance.2)The linear potential comes only from

exp(2πi(δ??1k,S)).The surface independence

of the static potential is assured due to the4-d

linking number.

3.A new gauge?xing method for abelian

projection

To con?rm the above results numerically,we

analyze gauge dependence of abelian projection

3 by Langevin equation with stochastic gauge?xing

term[12],

?

δA aμ(x,τ)+ 1

?A aμ?τ+ηaμ(x,τ)

√