color confinement in coulomb gauge qcd and color-dependent interactions

東京大学ハドロン研究室セミナー

Color confinement in Coulomb Color confinement in Coulomb gauge QCD and color-gauge QCD and color-dependent interactionsdependent interactions

Takuya Saito

斎藤卓也

共同研究者：中村純（広大）、土岐博（ RCNP) 、中川義之（ RCNP) 、 D. Zwanziger (NY)

Collaborators ： A.Nakamura （ Hiroshima ） ,H.Toki （ RCNP),Y.Nakagawa （ RCNP),D. Zwanziger (NY)


Part1: Part1: Study of color confinement scenario in Coulomb Study of color confinement scenario in Coulomb gauge: lattice calculation of color-Coulomb gauge: lattice calculation of color-Coulomb Instantaneous potential in color singlet channel~Instantaneous potential in color singlet channel~

Part2: Lattice study on color-dependent potentialPart2: Lattice study on color-dependent potentials of QCD; lattice study of the color 3s of QCD; lattice study of the color 3** quark-quark quark-quark potential, and 8 quark-antiquark, 6 qq potentials.potential, and 8 quark-antiquark, 6 qq potentials.


Study of color confinement Study of color confinement scenario in Coulomb gaugescenario in Coulomb gauge

~ lattice calculation of color-Coulomb ~ lattice calculation of color-Coulomb instantaneous potential ~instantaneous potential ~

Takuya Saito （ RCNP at Osaka Univ.）Collaborators ：

Y. Nakagawa （ RCNP at Osaka Univ. ）

H. Toki （ RCNP at Osaka Univ.)

A. Nakamura (RIISE at Hiroshima Univ.)

D. Zwanziger ( NY Univ.)


1. Motivation

2. Color confinement scenario in the Coulomb gauge Q

CD

3. Method ( partial-length Polyakov line )

4. Numerical results

（ in the confinement and deconfinement phases ）

5. Summary

ContentsContents


Confinement Confinement 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.

0T

0T

( ) , 0,at 0A

V R KR K TR

e( ) , 0,at 0

DM R

DV R M TR


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.

There were several approaches and a lot of works to understand the confinement …. :

In this study, we focus the Coulomb gauge QCD, and we will investigate the confinement mechanism in Coulomb gauge by the lattice QCD simulation.


Confinement scenario of CConfinement scenario of Coulomb gauge QCDoulomb gauge QCD

（ By Zwanziger ）

D. Zwanziger, PTP Suppl. No. 131, 233(1998); A.Cucchieri, D.Zwanziger, PRD65,014001,(2002). PRD65,014002,(2002)

1. Coulomb instantaneous potential in QCD

2. Difference between Wilson-loop and instantaneous potentials

3. FP-ghost operator and instantaneous potentials

4. Related topics for Coulomb gauge


　 Hamiltonian in the Coulomb gauge QCD

　 Faddeev-Popov term in the Coulomb gauge QCD

　 Time-time component of the gluon propagators.

Coulomb gauge QCDCoulomb gauge QCD

3 2 2 3 31 1( ) ( , ) ( )

2 2i iH d x E B d xd y x D x y y

3 21 1( , ) ( )

( , ) ( , )zD x y d zM x y M x y

2( )M gA

20 0( ) ( ) ( ) ( )g A x A y V x y P x y

24 4( ) ( , ) ( )V x y g D x y x y

Instantaneous part

retarded (vacuum polarization) part


Color-Coulomb instantaneous partColor-Coulomb instantaneous part Important quantity in the Coulomb gauge confinement

scenario

00 0 0( , ) ( , ) (0,0)D x t A x t A ( ) ( ) ( , ), | |coulV r t P x t r x

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.


Quark Wilson loop potential and color-Quark Wilson loop potential and color-Coulomb instantaneous potentialCoulomb instantaneous potential

Quark Wilson loop potentail, Vw ,should be distinguished from color-Coulomb instantaneous potentail Vc.

Color-Coulomb, Ｖｃ , is responsible for confinement.



Zwanziger’s inequality

( ) ( )phys coulV R V R

Zwanziger, PRL90, 102001 (2003)

If the physical potential is confining, then the color-Coulomb potential is also confining.

Here the physical potential corresponds to the Wilson loop potential.


Fadeev-Popov and instant. partsFadeev-Popov and instant. parts


Instantaneous part is defined in terms of FP operator in QCD

0 ; Gribov regionM 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.

21 1( ) , M: Fadeev-Popov operatorcoul iV r

M M


Related refs. for the Coulomb Related refs. for the Coulomb gauge QCD (1)gauge QCD (1)

1. Study of confinement by Gribov. NPB139,1 (1978)

2. Color-Coulomb instantaneous part is very important, which is advocated by Zwanziger, NPB518,237 (1998)

3. Study of the renormalization of the Coulomb gauge QCD, Baulieu, Zwanziger, NPB548,527(1998)

4. 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) )

5. There is an inequality, Vphys <=Vcoul, which is found by Zwanziger, PRL90, 102001 (2003)

6. 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). )


Related refs. for the Coulomb Related refs. for the Coulomb gauge QCD (2)gauge QCD (2)

7. 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.)

8. 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)


Aim in this studyAim in this study

By the SU(3) lattice simulation, we study the behavior of the colo

r-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.


MethodMethod1. Quantizaion by lattice regularizaion

2. Gauge fixing on lattice gauge theory

3. Measurement ( partial-length polyakov loops )


Lattice regularization

ˆˆ, , , ,1 Regauge x x x xS Tr U U U U

2Ncg2

U x, exp(igaA ( x))

1

4d 4 xTrF

2 , a 0

a

1

a

　Wilson action　　　　　　　

　 cut-off

　 link variable

　 Lattice regularization

( )

( )

S U

S U

DUOeO

DUe

　Path-integral quantization


Lattice regularization

　Expectation value and Monte Carlo method

( )

( )

S U

S U

DUOeO

DUe

Expectation values we want

Gauge configurations are generated by the probability

( )( ) S UP U e

1( )

N

kk

O O UN

After N times repeated, one can obtain physical quantities

0 1 { , , , }NU U U


Gauge fixing on a lattice

In general, a gauge fixing is not required in finite size lattices.

Iterative method to fix gauge confs.

Monte Carlo Steps

i iA

0i iA

Wilson-Mandula Method 　

PLB185,127(1987)

Gauge rotation a † ˆTr ( ) ( ) 0i ii

U x U x i

( ) 0, 0i ii

A x a

x,i

Maximize ReTrU ( )i x

† ˆ( ) ( ) ( ) ( ) ( )i i iU x U x x U x x i

( )( ) i ii A xx e † ( )x

( )iU x

ˆ( )x i


Measurement In this study, the most important issue is to extract the

instantaneous part from the gluon propagators. PRD67,094503(2003),PRD69,074506(2004).

0 t1

( , ) ( , ), T=1,2, NT

t

L x T U x t

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.

q qR

1T

2T

3T

tT N

( ,0)V R

Partial-length Polyakov line


Simulation parameters

One plaquette Wilson gauge action and quenched sim.

Lattices at zero temp. ： β=5.85-6.40, 184, 183x32, ３ 00

confs.

Lattices at finite temp.: β=6.11~7.0, 243x6, 300 confs.

A la Mandula-Oglive method for gauge fixing （ maximiza

tion of ReTrU ）

Computer facilities : ＮＥＣ　ＳＸ５　 of RCNP at Osak

a Univ.


Numerical results:Numerical results:(1)(1)

for for the confining the confining phasephase


Color-Coulomb potential （ confining phase）

( , ) / , A=-12

V R T C KR A R

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.

PTP115(2006)189-200

( ) ( )phys coulV R V R

instantaneousretarded (vacuum)


12 20 0 0

1

2 220 0 0( ) ( )

b

b b gLQCDa f g b g e

Scaling of Coulomb string tension

QCDC : String tension [MeV]: QCD mass scale [MeV]QCD

Asymptotic scaling

0 0( ) ( )LQCD

a K

f g f g

Beta function


Scaling of the color-Coulomb string tension

Color-Coulomb string tension scales monotonically as the lattice cutoff or the coupling constant.

C

If the asymptotic scaling of QCD is satisfied enough, then we will find the following relation:


Numerical results:Numerical results:(2)(2)

for for the deconfining pthe deconfining phasehase


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


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.


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.

( , ) R S si dx A RSW R S e e

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.


Temp. dep. of the spatial string tension

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,

2( ) ( )s mT C g T T

Here, let’s assume that the instantaneous part also satisfies the magnetic scaling.

G.S. Bali, et. al, PRL71,3059(1993)

This assumption is good for the data over T/Tc=2.


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


T dep. of instantaneous string tension

2

1 1

( )( )i

T

C g TT

102

0

12 ln ln 2ln

( )

bT Tb

g T b

10

0

2 ln ln 2lnc c

c c

T TbT Tb

T b T

, : free parameterC

Fitting function


T dep. of instantaneous string tension

2

1 1,

( )( )i

T

C g TT

Two-parameter fit ( T/Tc=2-4 )20.735(18), / 4.41(29), / 1.47cC T ndf

Spatial Wilson loop; two-parameter fit, ( NPB469 1996 410-444 )20.566(13), / 9.6(8), / ?cC T ndf

Spatial gluon propagator ( PRD69,014506,2004 )

0.486(31) 0.549(16)C If we use the electric scaling… ( T/Tc = 2-4 )

20.829(10), / 1.44(4), / 1.25cC T ndf It may be less proper since leading order perturbation gives C=1.


SummarySummary


SummarySummary We have investigated the behavior of the color-Coulomb in

stantaneous 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.


SummarySummary 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.


Future work

Color-Coulomb instantaneous potential is very closely related to the sing

ularity 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. (althou

gh we have no idea yet.)

Calculation of the color-dependent potential among two or three quarks p

otential.

Investigate of the non-instantaneous vacuum polarization ( retarded ) part

s. It may relate to the QGP phase transition, the chiral symmetry breaking, t

he pair quark creation, etc.


Takuya Saito

in collaboration with A. Nakamura

Lattice study onLattice study oncolor-dependent potentials of QCDcolor-dependent potentials of QCD

This presentation is based on PLB621(2005)171,PTP111(2004)733,PTP112(2004)183

and

in collaboration with H. Toki and Y. Nakagawa


1. QCD color quark potential

2. Polyakov loop correlator

3. Numerical results

4. Summary

Contents


IntroductionIntroduction

Color potentials in QCDColor 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.

　 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

8133 3633

3 3 3 1 8 8 10

: mesonQQ

QQ

baryonQQQ:


Singlet potential 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 !

0T

0T

1

qqV


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 !


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.


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.


Our aim in this study　 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.

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.


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 ?

Our aim in this study


Polyakov loop correlPolyakov loop correlatorsators

1. Polyakov line

2. Polyakov line correlator

3. Potentials between two quarks

4. Partial-Polyakov line correlator


Polyakov line

Polyakov line

Order parameter in pure gauge theory

00( , ) exp ' ( , ') ( ,0)

t a ax t T i dt t A x t x

0

1( , ) ( , ) 0a at A x t x t

i t

( ) ( ,0)L x x

0 0 0( ) ( , ) ( , 1)... ( ,1)t tL x U x N U x N U x

( , )x t

( ,0)x

( , )t tU x N

( ,1)tU x

0 , confinementTrL

0 , deconfinementq qF T

q

Fe

F

( McLerran, RMP58, 1021(1986) )


Polyakov line correlator

• Color singlet channel

11 2( )

1 2Tr ( ) ( )q qF x x

e L x L x

1( , )x t

1( ,0)x

2( , )x t

2( ,0)x

t

1 2R x x


( Nadkarni, PRD33,3738 )

21 ( 1)N N N

811 8

VV

qqG e P e P

1 ( ) †1e TrL(R)L (0)

3V R

Quark-antiquark potential

Color decomposition in quark-antiquark for SU(N)

Quark-antiquark correlator is made by the singlet and octet parts.

8 ( ) 8 3e TrL(R)TrL (0) TrL(R)L (0)

9 8V R

Singlet and octet potentials are defined by the Polyakov line correlator for SU(3)


( ) 3 1e TrL(R)TrL(0) TrL(R)L(0)

4 4symV R

( ) 3 1e TrL(R)TrL(0) TrL(R)L(0)

2 2anti symV R

1 1( 1) ( 1)

2 2N N N N N N

Quark-quark potentialS. Nadkarni, PRD34,3904

　 Color decomposition in quark-quark potential.

　 Symmetric and antisymmetric potentials are defined as

　 qq correlator is made by the following two parts

sym antisymV Vqq sym antisymG e P e P


Partial-length Polyakov line correlator

Here, the temporal extension is restricted. We can calculate PPL corr

elators in quenched lattice in the confinement region. Greensite, Olejnik, PRD67,094503(2003),PRD69,074506(2004).

†1( , ) ( , ) (0, ) , R= x

3G R T Tr L R T L T

( , )( , ) log

( , 1)

G R TV R T

G R T

( ,0) log ( ,1)V R G R

V(R,0) corresponds to the color-Coulomb instantaneous potential, Vcoul(R).

q qR

1T

2T

3T

tT N

( ,0)V R


Numerical resultsNumerical results

Color-dependent forces Color-dependent forces between two quarksbetween two quarks


Typical behavior for 4 color-dependent potentials

and Casimir scaling


Color-dep. potentials between two quarks

1. Singlet and antisymmetric pots. are linearly rising pots. for large quark separation. They can be described by the Coulomb and linear terms.

2. The distance dependence in the repulsive channel seems to be complicated, and this result is not conclusive. More extensive simulation is required.

a~0.124fm

A. Nakamura, T. Saito

PLB621(2005),171-175

T=0

( ) /V R c KR A R


1. String tensions are described in terms of the Casimir.

Casimir scaling

Coulomb gauge

Ratio of the Casimir between 1 and 3*

*

1

3

4 / 32

2 / 3

C

C

A. Nakamura, T. Saito

PLB621(2005),171-175


Example of the behavior for 4 color-dependent potentials

in the QGP phase


Color-dep. potential in QGP phase

A.Nakamura, T.Saito

PTP111(2004)733-743

PTP112(2004)183-188

We obtain the screened potentials in each color channel in the QGP phase.

Landau gauge


SummarySummary

and and

future worksfuture works


SummarySummary

1. We have calculated the two-quark potentials in each col

or channel with Polyakov line correlator in the hadron (Q

GP) phase.

2. Quark-quark antisymmetric 3* potential is a linearly risin

g potential, and we checked the Casimir scaling.

3. In our calculation, it is not conclusive for the long-distan

ce behavior in the repulsive channels.

4. The potentials in each color channel are color-screened

in the QGP phase.


Future worksFuture worksColor-octet and color-symmetric channels may be requir

ed more extensive lattice studies, to get the conclusive res

ult.

Divergence of a color flux in color non-singlet channel.

Calculation of 3-quark potentials, and the behavior of the

2-quark potentials in the 3-quark potential.

Dynamical quark simulations; it may be easier than quen

ched simulations, because the expectation value of TrL do

es not vanish even in the confinement phase. (not possing

Z(3) symmetry.)

color confinement in coulomb gauge qcd and color-dependent interactions

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