Transcript Confinement scenario of Coulomb gauge QCD

Color confinement in Coulomb
gauge QCD and colordependent interactions
Takuya Saito
斎藤卓也
Collaborators：A.Nakamura（Hiroshima）,H.Toki
（RCNP),Y.Nakagawa（RCNP),D. Zwanziger (NY)
共同研究者：中村純（広大）、土岐博（RCNP)、中川義之（RCNP)、
D. Zwanziger (NY)
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Part1:
Study of color confinement scenario in Coulomb
gauge: lattice calculation of color-Coulomb
Instantaneous potential in color singlet channel~
Part2: Lattice study on color-dependent
potentials of QCD; lattice study of the color 3*
quark-quark potential, and 8 quark-antiquark, 6
qq potentials.
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Study of color confinement
scenario in Coulomb gauge
~ lattice calculation of color-Coulomb
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.)
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Contents
1.
Motivation
2.
Color confinement scenario in the Coulomb gauge
QCD
3.
Method ( partial-length Polyakov line )
4.
Numerical results
（in the confinement and deconfinement phases）
5.
Summary
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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.
T 0
In QCD lattice simulation, the quark
potential rises linearly for the large
quark separation, implying the nonvanishing string tension.
However, there is a problem how
QCD produces the confinement of the
quarks and gluons.
T 0
V ( R)
V ( R)
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A
KR  , K  0, at T  0
R
e M D R
, M D  0, at T  0
R
Confinement
There were several approaches and a lot of works to understand the
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.
In this study, we focus the Coulomb gauge QCD,
and we will investigate the confinement
mechanism in Coulomb gauge by the lattice QCD
simulation.
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Confinement scenario of
Coulomb gauge QCD
（By Zwanziger）
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
D. Zwanziger, PTP Suppl. No. 131, 233(1998);
A.Cucchieri, D.Zwanziger, PRD65,014001,(2002).
PRD65,014002,(2002)
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Coulomb gauge QCD
Hamiltonian in the Coulomb gauge QCD
H


1 3
1 3 3
2
2
d
x
E

B

d xd y   ( x ) D ( x, y )  ( y ) 
i
i


2
2
Faddeev-Popov term in the Coulomb gauge QCD
 1
1 
2
D( x, y)   d z 
( z )
M  (2  gA)

M ( x, y) 
 M ( x, y)
3
Time-time component of the gluon propagators.
retarded (vacuum
polarization) part
g 2 A0 ( x) A0 ( y)  V ( x  y)  P( x  y)
Instantaneous part
V ( x  y)  g 2 D( x, y)  ( x4  y4 )
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Color-Coulomb instantaneous part
Important quantity in the Coulomb gauge confinement
scenario
D00 ( x, t )  A0 ( x, t ) A0 (0,0) 
 Vcoul (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.
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Quark Wilson loop potential and colorCoulomb instantaneous potential
D00 ( x, t )  A0 ( x, t ) A0 (0,0) 
 Vcoul (r ) (t )  P( x, t ), r | x |
Quark Wilson loop potentail, Vw ,should be
distinguished from color-Coulomb instantaneous
potentail Vc.
Color-Coulomb, Ｖｃ, is responsible for confinement.
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Zwanziger’s inequality
Zwanziger, PRL90, 102001 (2003)
Vphys (R)  Vcoul (R)
Here the physical potential corresponds to the
Wilson loop potential.
If the physical potential is confining, then the
color-Coulomb potential is also confining.
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Fadeev-Popov and instant. parts
D00 ( x, t )  A0 ( x, t ) A0 (0,0) 
 Vcoul (r ) (t )  P( x, t ), r | x |
Instantaneous part is defined in terms of FP operator in QCD


1
1
2
Vcoul (r ) 
i
, M: Fadeev-Popov operator
M
M
M  0 ; Gribov region
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.
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Related refs. for the Coulomb
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). )
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Related refs. for the Coulomb
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 nonvanishing 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)
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Aim in this study
By the SU(3) lattice simulation, we study the behavior of the
color-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 nonvanishing string tension.
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Method
1. Quantizaion by lattice regularizaion
2. Gauge fixing on lattice gauge theory
3. Measurement ( partial-length polyakov loops )
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Lattice regularization
Lattice regularization

a
cut-off
a
1

U x,  exp( igaA ( x))
link variable
Wilson action

S gauge    1  Re Tr U x , U x  ˆ , U xˆ , U x, 

Path-integral quantization

2N c
g
2
O 
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
1 4
2
d
xTrF
, a0

4
 S (U )
DUOe

 S (U )
DUe


Lattice regularization
Expectation value and Monte Carlo method
Expectation values we want
O 
 S (U )
DUOe

 S (U )
DUe

Gauge configurations are generated by the probability
P(U ) e S (U )
 { U0 ,U1,
,U N }
After N times repeated, one can obtain physical quantities
1 N
O   O(U k )
N k
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Gauge fixing on a lattice
 In general, a gauge fixing is not required in finite size
lattices.
i A ( x )
( x)  e
 Iterative method to fix gauge confs.
† ( x)
Ui ( x) Ui ( x)  † ( x)Ui ( x)( x  iˆ)
Maximize  ReTrUi ( x)

x,i

i Ai
i i
( x  iˆ)
Ui ( x)
Monte Carlo Steps

a
†
ˆ
Tr

U

i ( x)  Ui ( x  i )  0
Gauge rotation
i
i Ai  0
  A ( x)  0, a  0
i
Wilson-Mandula Method
i
i
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PLB185,127(1987)
Measurement
In this study, the most important issue is to extract the
instantaneous part from the gluon propagators.
PRD67,094503(2003),PRD69,074506(2004).
Partial-length Polyakov line
T  Nt
T
L( x, T )  U 0 ( x, t ), T=1,2,
t 1
T 3
T 2
V ( R, 0)
T 1
q
R
q
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.
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Nt
Simulation parameters
One plaquette Wilson gauge action and quenched sim.
Lattices at zero temp.：β=5.85-6.40, 184, 183x32, ３00
confs.
Lattices at finite temp.: β=6.11~7.0, 243x6, 300 confs.
A la Mandula-Oglive method for gauge fixing
（maximization of ReTrU）
Computer facilities : ＮＥＣ ＳＸ５ of RCNP at Osaka Univ.
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Numerical results:
(1)
for the confining
phase
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Color-Coulomb potential （confining phase）
V(R,0) is a linearly rising
potential, i.e., confining
potential.
instantaneous
retarded (vacuum)
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.
V ( R, T )  C  KR  A / R, A=-

12
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Vphys (R)  Vcoul (R)
PTP115(2006)189-200
Scaling of Coulomb string tension
Asymptotic scaling
  CQCD
 : String tension [MeV]
QCD : QCD mass scale [MeV]
Beta function
a LQCD  f ( g 0 )  (b0 g )
2
0

 LQCD

b1
2 b02
a

e
1
2 b0 g 02
K


f ( g0 ) f ( g0 )
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Scaling of the color-Coulomb string tension
 If the asymptotic scaling
of QCD is satisfied enough,
then we will find the
following relation:


C
 Color-Coulomb string
tension scales
monotonically as the lattice
cutoff or the coupling
constant.
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Numerical results:
(2)
for the deconfining
phase
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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 Yukawatype potenial.
PTP115(2006)189-200
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Color-Coulomb potential（deconfining
phase) : at higher temperature
 Linearity of
instantaneous part dose not
vanish at high temperature.
 Appearance of any nonperturbative mode !?
 Instantaneous part , not
having explicitly the time
variable, may not be
sensitive to time
(temperature) variable.
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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.
dx

RS
W ( R, S )  e
i

A
e  s RS
 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.
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Temp. dep. of the spatial string tension
G.S. Bali, et. al, PRL71,3059(1993)
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,
 s (T )  Cm g (T )T
2
This assumption is good for the data over
T/Tc=2.
Here, let’s assume that the instantaneous
part also satisfies the magnetic scaling.
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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).
log scale
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 In any cases, it is clear
that there exist the colorCoulomb string tensions
after the QGP phase
transition, which are scaled
with the temperature.
T dep. of instantaneous string tension
Fitting function
T
1 1

2
C
g
(T )
 i (T )
C,   : free parameter
 T
1
T b1 
 2b0 ln
 ln  2ln 
2
g (T )
 b0 
 

 

 T Tc  b1 
 T Tc  
 2b0 ln 
  ln  2ln 
 
 Tc   b0 
 Tc   
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T dep. of instantaneous string tension
T
1 1

,
2
 i (T ) C g (T )
Two-parameter fit ( T/Tc=2-4 )
C  0.735(18), Tc /   4.41(29),  2 / ndf  1.47
Spatial Wilson loop; two-parameter fit, ( NPB469 1996 410-444 )
C  0.566(13), Tc /   9.6(8),  2 / ndf  ?
Spatial gluon propagator ( PRD69,014506,2004 )
C  0.486(31)  0.549(16)
If we use the electric scaling… ( T/Tc = 2-4 )
C  0.829(10), Tc /   1.44(4),  2 / ndf  1.25
It may be less proper since leading order perturbation gives C=1.
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Summary
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Summary
We have investigated the behavior of the color-Coulomb
instantaneous 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 colorCoulomb string tension in the confinement phase, while in the
deconfinement phase, it produces the screened potential.
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Summary
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.
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Future work
Color-Coulomb instantaneous potential is very closely related to the
singularity 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.
(although we have no idea yet.)
Calculation of the color-dependent potential among two or three quarks
potential.
Investigate of the non-instantaneous vacuum polarization ( retarded )
parts. It may relate to the QGP phase transition, the chiral symmetry
breaking, the pair quark creation, etc.
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Lattice study on
color-dependent potentials of QCD
Takuya Saito
in collaboration with A. Nakamura
This presentation is based on
PLB621(2005)171,PTP111(2004)733,PTP112(2004)183
and
in collaboration with H. Toki and Y. Nakagawa
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Contents
1. QCD color quark potential
2. Polyakov loop correlator
3. Numerical results
4. Summary
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Introduction
Color potentials in QCD
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Color potentials in QCD
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
QQ: meson
QQ
QQQ: baryon
3  3  1 8
33  6 3
3  3  3  1  8  8  10
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.
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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
1
qq
V
T 0
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 !
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T 0
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 !
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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.
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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.
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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 colordependent 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 colordependent potential by lattice QCD simulation.
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Our aim in this study
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 ?
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Polyakov loop
correlators
1. Polyakov line
2. Polyakov line correlator
3. Potentials between two quarks
4. Partial-Polyakov line correlator
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Polyakov line
Polyakov line
( McLerran, RMP58, 1021(1986) )
1 

a a

t
A
(
x
,
t
)
0
 i t
 ( x , t )  0



t
 ( x, t )
U t ( x , Nt )

 ( x , t )  T exp i  dt 't a A0a ( x , t ')  ( x ,0)
0
L( x ) ( x ,0)
L( x )  U 0 ( x , Nt )U 0 ( x , N t  1)...U 0 ( x ,1)
Order parameter in pure gauge theory
TrL
e
 FqT
  0  Fq  , confinement

 0  Fq  , deconfinement
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U t ( x ,1)
 ( x ,0)
Polyakov line correlator
Two-quark state at t=0
|    ( x,0) ( ) ( x,0) | s 
†
a
c b
†
 ( x2 , t )
 ( x1 , t )
Quark-antiquark potential
e
  Fqq
e
  Fqq
 H


|
e
| 


a
c b

s

|
(
x
,0)(

) ( x2 , 0)

1
a ,b , s
 H
a
   s |e
 H
e
 ( x1 ,0) ( ) ( x2 , 0) | s 
†
 ( x1 ,  ) ( x1 ,0)
a
†
c b
a
†
R  x1  x2
 ( x1 ,0)
 ( x 2 ,0)
t
a ,b , s
( c )b ( x2 ,  )( c )b ( x2 ,0) † | s 
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Polyakov line correlator
• Color singlet channel
e
  Fq1q ( x1  x2 )
 ( x2 , t )
 ( x1 , t )
TrL( x1 ) L ( x 2 )
R  x1  x2
 ( x1 ,0)
 ( x 2 ,0)
t
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Quark-antiquark potential
( Nadkarni, PRD33,3738 )
Color decomposition in quark-antiquark for SU(N)
N  N  1 ( N 2 1)
Quark-antiquark correlator is made by the singlet and octet parts.
Gqq  eV1 P1  eV8 P8
Singlet and octet potentials are defined by the Polyakov line correlator
for SU(3)
e
V1 ( R )
e V8 ( R )
1
 TrL(R)L† (0)
3
8
3

 TrL(R)TrL (0)  TrL(R)L (0)
9
8
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Quark-quark potential
S. Nadkarni, PRD34,3904
Color decomposition in quark-quark potential.
1
1
N  N  N ( N  1)  N ( N  1)
2
2
qq correlator is made by the following two parts
Gqq  e
Vsym
Psym  e
Vantisym
Pantisym
Symmetric and antisymmetric potentials are defined as
e
e
Vsym ( R )
Vanti sym ( R )
3
1
 TrL(R)TrL(0)  TrL(R)L(0)
4
4
3
1
 TrL(R)TrL(0)  TrL(R)L(0)
2
2
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Partial-length Polyakov line correlator
Here, the temporal extension is restricted. We can calculate PPL
correlators in quenched lattice in the confinement region. Greensite,
Olejnik, PRD67,094503(2003),PRD69,074506(2004).
G ( R, T ) 
T  Nt
1
Tr  L( R, T ) L† (0, T )  , R= x
3
V ( R, T )  log
T 3
G( R, T )
G( R, T  1)
T 2
V ( R, 0)
T 1
V ( R,0)   log G( R,1)
q
R
q
V(R,0) corresponds to the color-Coulomb instantaneous potential, Vcoul(R).
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Numerical results
Color-dependent forces
between two quarks
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Typical behavior for 4 colordependent potentials
and
Casimir scaling
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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.
T=0
a~0.124fm
2. The distance dependence
in the repulsive channel
seems to be complicated,
and this result is not
conclusive. More
extensive simulation is
required.
V ( R)  c  KR  A / R
東京大学ハドロン研究室セミナー
A. Nakamura, T. Saito
PLB621(2005),171-175
Casimir scaling
Ratio of the Casimir
between 1 and 3*
C1 4 / 3

2
C3* 2 / 3
1. String tensions are
described in terms of
the Casimir.
Coulomb gauge
A. Nakamura, T. Saito
PLB621(2005),171-175
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Example of the behavior
for 4 color-dependent potentials
in the QGP phase
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Color-dep. potential in QGP phase
We obtain the screened
potentials in each color
channel in the QGP
phase.
Landau gauge
A.Nakamura, T.Saito
PTP111(2004)733-743
PTP112(2004)183-188
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Summary
and
future works
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Summary
1. We have calculated the two-quark potentials in each
color channel with Polyakov line correlator in the hadron
(QGP) phase.
2. Quark-quark antisymmetric 3* potential is a linearly
rising potential, and we checked the Casimir scaling.
3. In our calculation, it is not conclusive for the longdistance behavior in the repulsive channels.
4. The potentials in each color channel are color-screened
in the QGP phase.
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Future works
Color-octet and color-symmetric channels may be
required more extensive lattice studies, to get the
conclusive result.
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
quenched simulations, because the expectation value of
TrL does not vanish even in the confinement phase. (not
possing Z(3) symmetry.)
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